JP3799511B2

JP3799511B2 - Perspective 3D NURB curve automatic restoration method and apparatus

Info

Publication number: JP3799511B2
Application number: JP07014997A
Authority: JP
Inventors: 理金井; 明彦遠藤; 邦彦堀田
Original assignee: EXA CO Ltd
Current assignee: EXA CO Ltd
Priority date: 1997-03-24
Filing date: 1997-03-24
Publication date: 2006-07-19
Anticipated expiration: 2017-03-24
Also published as: JPH10269380A

Description

【０００１】
【発明の属する技術分野】
本発明は、コンピュータ支援３次元入力システムにおける３次元形状入力方式に関し、特にＣＡＩＤ（コンピュータ支援工業デザイン）／ＣＡＤ（コンピュータ支援設計）／ＣＧ（コンピュータグラフィックス）システムにおいて２次元の透視図から３次元曲線を復元し、立体形状として表示する３次元自動復元システムに関するものである。
【０００２】
【従来の技術】
デザイナーを含む操作者（つまり人間）は、個人差はあるものの、２次元上に描かれた透視図のスケッチを組み合わせて、自由曲線を含む３次元の立体形状を頭の中でイメージできる。
【０００３】
図８及び図９はそのような３次元の立体形状をイメージするための２次元の透視投影図である。図８は投影面１を示す透視投影図９０であり、図９は投影面２を示す透視投影図９２である。
【０００４】
従来、精密工学会誌、vol.59、no.6、1993、精密工学会、日本、古島、金井、高橋、「手書き図形の自動認識による３次元自由曲線モデルの生成」や、電子情報通信学会論文誌Ｄ、J71-D 、（1988）84、電子情報通信学会、日本、田中敏光ほか、「Generalized symmetryに基づく３次元形状の復元」に記載されているように、ＣＡＩＤ／ＣＡＤ／ＣＧシステム等で、このような２次元の透視投影図から３次元の立体形状を復元し、操作者の立体形状のイメージを支援するような３次元自動復元システムが開発されている。
【０００５】
このような従来の３次元自動復元システムにおいては、透視投影図に描画された２次元曲線を一度２次元の点列で表現し、１又は複数の透視投影図の点列を基にして３次元の点列に変換し、その３次元の点列に３次元曲線を当てはめて立体形状を復元させている。
【０００６】
【発明が解決しようとする課題】
しかし、上述のように、１つあるいは複数の透視投影図から３次元自由曲線を復元する場合、２次元自由曲線を２次元点列で表現しておき、１又は複数投影面上の２次元点列を基にして、ある３次元点列に変換し、その３次元の点列に３次元曲線を当てはめて立体形状を復元させていたため、工業製品で表現されるような自由度の高い立体形状の透視投影図は入力することができないか、または入力が困難であった。また、立体面上の円や円弧のような２次曲線を理論的に正確に表現することができなかった。
【０００７】
本発明は、上記のような問題点を解決するためになされたものであり、手書きの誤差を含んでいるような２次元自由曲線又は円や円弧を表現している２次曲線が入力されても、滑らかな３次元曲線あるいは、立体面上の滑らかな２次曲線として表現でき、簡単な入力で操作者のイメージに近い立体形状が得られるような３次元自動復元システムを得ることを目的とする。なお、ここでいう自由曲線とは直線等も含むものである。
【０００８】
【課題を解決するための手段】
本発明に係る透視図３次元ＮＵＲＢ曲線自動復元方法は、投影面決定手段が１又は複数の投影面を決定する工程と、決定した１又は複数の投影面に対し、その投影面に立体形状を投影した時の２次元形状を構成する２次元自由曲線が入力されると、３次元ＮＵＲＢ曲線変換手段が２次元自由曲線に基づいて３次元ＮＵＲＢ曲線を算出し、１又は複数の投影面の３次元ＮＵＲＢ曲線に基づいて立体形状を復元する工程とを有している。
本発明においては１又は複数の投影面を決定し、その投影面に立体形状を投影した時の２次元形状を構成する２次元自由曲線を手書き等により入力する。その２次元自由曲線に基づいて、描ける曲線に制限がなく、微小変化の影響が少ない３次元ＮＵＲＢ曲線を算出し、その３次元ＮＵＲＢ曲線に基づいて立体形状を復元する。
【０００９】
また、本発明に係る透視図３次元ＮＵＲＢ曲線自動復元方法の１又は複数の投影面は、入力手段を介して入力される、表示手段が表示する３次元座標空間上の補助用直方体に対する回転移動又は平行移動指示に基づいて、投影面決定手段が決定する。
本発明においては、立体形状を復元する３次元座標空間上に表示された補助用直方体を回転移動又は平行移動させることにより投影面に対応した視点から補助用直方体を表示させることで視覚的に投影面を決定する。
【００１０】
また、本発明に係る透視図３次元ＮＵＲＢ曲線自動復元方法において３次元ＮＵＲＢ曲線変換手段が３次元ＮＵＲＢ曲線を算出する工程は、同一投影面上において面対称関係にある２次元自由曲線又は複数投影面上において立体形状の同一線を表すものと見なされる２次元自由曲線の組み合わせを抽出し、それぞれの２次元自由曲線をアフィン変換してそれぞれの端点の座標の補正を行って算出し、推定したそれぞれの点列の近傍を２次元ＮＵＲＢ曲線が通過する際の点列通過パラメータの値に基づいて２次元ＮＵＲＢ曲線の制御点の座標を算出し、２次元ＮＵＲＢ曲線の制御点及び透視変換行列に基づいて算出された３次元ＮＵＲＢ曲線の制御点の座標に基づいて３次元ＮＵＲＢ曲線を算出する。
本発明においては、同一投影面上において面対称関係にある２次元自由曲線又は複数投影面上において立体形状の同一線を表すものと見なされる２次元自由曲線の組み合わせを抽出し、それぞれの２次元自由曲線をアフィン変換により端点の座標の補正を行って算出する。またそれぞれの点列の近傍を２次元ＮＵＲＢ曲線が通過する際の点列通過パラメータの値を推定し、点列通過パラメータの値に基づいて２次元ＮＵＲＢ曲線の制御点の座標を算出し、２次元ＮＵＲＢ曲線の制御点及び透視変換行列に基づいて算出された３次元ＮＵＲＢ曲線の制御点の座標に基づいて３次元ＮＵＲＢ曲線を算出する。
【００１１】
また、本発明に係る透視図３次元ＮＵＲＢ曲線自動復元方法の点列通過パラメータは、点列間の距離の比例関係に基づいて設定される。本発明においては、それぞれの点列の近傍を２次元ＮＵＲＢ曲線が通過する際の点列通過パラメータは点列間の距離に比例して設定される。
【００１２】
また、本発明に係る透視図３次元ＮＵＲＢ曲線自動復元方法は、２次元ＮＵＲＢ曲線の制御点の座標は、２次元ＮＵＲＢ曲線の制御点を変数とする方程式を算出し、方程式、２次元自由曲線の端点の座標、点列の座標及び点列通過パラメータの値に基づいて、点列との誤差が最小となるような変数の値を２次元ＮＵＲＢ曲線の制御点の座標として決定する。
本発明においては、２次元ＮＵＲＢ曲線の制御点を変数とする方程式を算出する。その方程式に２次元自由曲線の端点の座標、点列の座標及び点列通過パラメータの値によって得られる制御点間の関係式のうち点列との誤差が最小となるような変数の値を２次元ＮＵＲＢ曲線の制御点の座標として決定する。
【００１３】
また、本発明に係る透視図３次元ＮＵＲＢ曲線自動復元方法の３次元ＮＵＲＢ曲線の制御点の座標の算出は、透視変換行列に基づいた２次元ＮＵＲＢ曲線の制御点と３次元ＮＵＲＢ曲線の制御点の座標との関係式を算出し、複数の投影面間で同一の制御点の関係を表す関係式において、誤差が最小になるように３次元座標に関し関係式を解き、３次元ＮＵＲＢ曲線の制御点の座標を算出する。
本発明においては、透視変換行列に基づいた２次元ＮＵＲＢ曲線の制御点と３次元ＮＵＲＢ曲線の制御点の座標との関係式を算出する。複数の投影面間で同一の制御点の関係を表す関係式において、誤差が最小になるように３次元座標に関し関係式を解き、複数の投影面間の手書き誤差を最小限に抑えた３次元ＮＵＲＢ曲線の制御点の座標を算出する。
【００１４】
また、本発明に係る透視図３次元ＮＵＲＢ曲線自動復元方法において３次元ＮＵＲＢ曲線変換手段が３次元ＮＵＲＢ曲線を算出する工程は、更に透視変換行列に基づいて、算出された３次元ＮＵＲＢ曲線の制御点の座標を１又は複数の投影面に再投影し、再投影により得られる２次元ＮＵＲＢ曲線の制御点の座標を新たな２次元ＮＵＲＢ曲線の制御点の座標として点列通過パラメータを再推定し、点列と、算出された２次元ＮＵＲＢ曲線との距離の関係の変化率が収束したものと判断するまで２次元ＮＵＲＢ曲線の制御点の座標及び点列のパラメータを再推定し、前記３次元ＮＵＲＢ曲線を算出する。
本発明においては、算出された３次元ＮＵＲＢ曲線の制御点の座標を１又は複数の投影面に再投影する。誤差が最小限に抑えられた３次元ＮＵＲＢ曲線の制御点を２次元ＮＵＲＢ曲線の制御点に変換し、それをあらたな２次元ＮＵＲＢ曲線の制御点として点列通過パラメータを再推定し、それらに基づいて点列と、算出された２次元ＮＵＲＢ曲線との距離の関係の変化率が収束し、誤差が抑えられるまで２次元ＮＵＲＢ曲線の制御点の座標及び点列のパラメータを再推定し、３次元ＮＵＲＢ曲線を算出する。
【００１５】
【発明の実施の形態】
図１は本発明の第１の実施の形態に係る透視図３次元ＮＵＲＢ曲線自動復元方法を実施するためのブロック図である。図において、２０１はコマンド解析／処理部であり、入出力機構２０５を介して表示装置２０６、スケッチ入力手段２０９と接続され、入力されるコマンドの解析及び処理を行う。２０２は幾何処理部であり、点の座標値の集合で与えられた２次元自由曲線の座標値に基づいて、立体形状を復元する。２０３は記憶処理部であり、図形記憶領域２０４にデータを記憶するための処理を行う。２０５は入出力機構であり、コマンド解析／処理部２０１、幾何処理部２０２、記憶処理部２０３及び図形情報記憶領域２０４により構築された部分と表示装置２０６及びスケッチ入力手段２０９とのインターフェイスとなる部分である。スケッチ入力手段２０９は操作者の操作が入力される部分でありマウス２０７及びキーボード２０８で構成されている。
【００１６】
図２は図１のブロック図を実際にハード構成する場合の構成図である。透視図３次元ＮＵＲＢ曲線自動復元方法は、主としてコンピュータ上で復元処理される。図２では中央計算機１０１、主記憶装置１０２、入出力制御装置１０３、マウス又はタブレット１０４、キーボード１０５、外部記憶装置１０６及び表示装置１０７をシステムバス１０８（又はこれに類似の機構）で接続した計算機システム１００で構成されている。またこれに類似するような計算機システムで構成される場合もある。
【００１７】
図３は幾何処理部２０２の構成を示すブロック図である。図において、５０は投影面決定手段５０であり、投影面決定の補助とするような直方体を表示装置２０６に表示させるようにしておき、スケッチ入力手段からの回転又は移動指示により直方体の視点を変化表示させて操作者が視点を決定すると、その視点の透視変換行列を算出して出力する。５２は同一投影図組合せ検出手段であり、あるひとつの透視投影図の２次元手書き曲線において、面対称な関係にある頂点（曲線の始点及び終点）・２次元曲線の組み合わせを検出する。５４は同一投影図頂点算出手段であり、同一投影図組合せ手段５２により検出された頂点の３次元座標位置を算出する。５６は複数投影図組合せ手段であり、複数の透視投影図の２次元手書き曲線において、空間内で同一頂点・曲線として対応づけられる頂点（曲線の始点及び終点）・２次元自由曲線の組み合わせを検出する。５８は複数投影図頂点算出手段であり、複数投影図組合せ手段５６により検出された頂点の３次元座標位置を算出する。６０は３次元ＮＵＲＢ曲線変換手段であり、同一投影図組合せ検出手段５２又は複数投影図組合せ検出手段５４で検出された２次元ＮＵＲＢ曲線、同一投影図頂点算出手段５６又は複数投影図頂点算出手段５８において算出された頂点座標、投影面上の入力点列座標及び投影面の透視変換行列に基づいて、アフィン変換（ｘ軸又はｙ軸方向の拡大縮小、変形等）を行って頂点や曲線を修正しながら２次元自由曲線から３次元ＮＵＲＢ曲線を復元する。
【００１８】
ここで、ＮＵＲＢ曲線について説明する。ＮＵＲＢ曲線とはNonUniform Rati-onal B-Spline 曲線のことである。平面上の曲線をパラメータｔを用いてパラメトリックに表現すると、例えば３次の多項式によるパラメータ曲線は、次式（１）のようになる。
Ｒ（ｔ）＝ｔ³Ｐ₃＋ｔ²Ｐ₂＋ｔＰ₁＋Ｐ₀ …（１）
となるここでＰ₃、Ｐ₂、Ｐ₁及びＰ₀はベクトルである。ベジエ曲線やB-Spline曲線はこのパラメトリック曲線形式で表現されるが、ベジエ曲線は制御点と呼ばれる点によって曲線が決定され、B-Spline曲線は曲線間の接続を制御点とは独立で行うことができる。
【００１９】
これらのパラメトリック曲線は正確に表現できる形状には制限があり、正確に表現する方法の１つに有理パラメトリック曲線と呼ばれる有理の多項式の形式で表された表現がある。ＮＵＲＢ曲線は有理パラメトリック曲線の１つであり、次式（２）で表される。
【００２０】
【数１】

【００２１】
ここでｗ_iが制御点Ｐ_iに対応する重みであり、Ｎ_i,p（ｔ）はB-Spline関数と同じである。ｐは曲線セグメントの次数である。これがＮＵＲＢ曲線である。
【００２２】
図４は自由度の高い曲線や円等のプリミティブな曲線を生成する入出力動作と幾何処理部２０２との関係を示す図である。この図は図８及び図９の透視投影図に基づいている。図３、図４並びに図８及び図９に基づいて幾何処理部２０２について詳しく説明する。
【００２３】
操作者は、スケッチ入力手段２０９より投影面１を２次元曲線の描画方向の視点として指定する（操作１）。この投影面１には、面対称であると考えられる２次元自由曲線が含まれている。このような曲線が含まれた投影面が最初に選ばれるようにする。投影面の指定には、投影面決定手段５０により、補助となる直方体をあらかじめ表示手段２０６に表示させるようにしておき、この直方体を操作者がスケッチ入力手段２０９を介して回転又は移動させることにより決定させるようにしておく。
【００２４】
操作者は投影面１を描画方向の視点として指定すると、２次元曲線の手書き入力を行う。曲線１が入力されても、組み合わせ対象がないため、同一投影図組合せ検出手段５２は処理を行わない。また複数投影図組合せ手段５６も投影面が１つしかないため、この時点では処理を行わない。曲線２、３及び４が入力されると、同一投影図組合せ検出手段５２は曲線１との組み合わせ対象が入力されたことを検出する。検出された曲線の端点（立体形状の頂点）に基づいて、同一投影図頂点算出手段５４は、投影面決定手段５０で決定された投影面に対応する透視変換行列Ｍⁱにより変換を行い、３次元上における曲線１〜曲線４の頂点の座標を決定する。座標が決定されると、曲線１〜曲線４上の点列をサンプリングする。必要なデータが３次元ＮＵＲＢ曲線変換手段６０に入力されて復元処理され、表示手段２０６に立体形状として表示させる（操作２）。手書きの誤差を含み、曲率が変化する自由度の高い曲線であるものの、ＮＵＲＢ曲線の性質上、微小変化の影響を受けないので、滑らかな３次元ＮＵＲＢ曲線を生成することができる。
【００２５】
次に操作者は曲線５の入力を行う。同一投影図組合せ検出手段５２は曲線５に対しては面対称の曲線が投影図９０上に存在しないとする。表示手段２０６に表示される立体形状はそのままである（操作３）。別の投影面からの同一曲線が複数投影図組合せ検出手段５６により認められると、３次元化される。
【００２６】
操作４では、２次元ＮＵＲＢ曲線による入力で、曲線６及び曲線７が楕円として入力される。この曲線６及び曲線７についても対応する曲線が投影面１上に存在ないので、表示手段２０６に表示される立体形状はそのままである（操作４）。曲線６及び曲線７についても別の投影面からの同一曲線が複数投影図組合せ検出手段５６により認められると、３次元化される。
【００２７】
曲線８を入力すると、同一投影図組合せ検出手段５２は曲線８に対しては対応する曲線が投影図９０上に存在しないとする。曲線９に対してはそれ自身が面対称となる曲線として存在すると判断して、同一投影図頂点算出手段５４は、透視変換行列Ｍⁱにより変換を行い、３次元上における曲線９の頂点の座標を決定し、曲線状の点列のサンプリングを行う。必要なデータが３次元ＮＵＲＢ曲線変換手段６０に入力され、３次元ＮＵＲＢ曲線の制御点を出力する。３次元形状表示制御手段６２は出力された３次元ＮＵＲＢ曲線の制御点に基づいて表示手段６４に立体形状を表示させる（操作５）。
【００２８】
投影面１からの手書きによる２次元曲線の入力が終了すると、再度、操作者は、スケッチ入力手段２０９を介して表示装置２０６に表示されている補助用の直方体を回転、移動させ、投影面２を２次元曲線の描画方向の視点として指定する（操作６）。ここで、現在立体形状における位置が決定している曲線１、２、３、４及び９に対応し、この位置の視点で見ることのできる曲線１１、１２、１９についてはすでに投影面２上に描かれている。したがってその曲線に基づいて他の曲線を入力していくことになる。
【００２９】
操作者により曲線１５が入力されると、同一投影面には組み合わせ対象がないため、同一投影図組合せ検出手段５２は処理を行わない。複数投影図組合せ手段５６は、投影面１及び２に基づいて、曲線５と曲線１５とが同一曲線を表すものと判断する。曲線５及び曲線１５に基づいて複数投影図頂点算出手段５８は頂点の座標を決定し、またそれぞれの曲線の点列の座標を算出して曲線５及び１５の２次元ＮＵＲＢ曲線を生成する。その後、必要なデータが３次元ＮＵＲＢ曲線変換手段６０に入力されて復元処理され、表示手段２０６に立体形状として表示させる（操作７）。
【００３０】
操作者により曲線１６及び１７が入力されると、同一投影面には組み合わせ対象がないため、同一投影図組合せ検出手段５２は処理を行わない。複数投影図組合せ手段５６は、投影面１及び２に基づいて、曲線６及び曲線１６並びに曲線７及び曲線１７とが同一曲線を表すものと判断する。曲線６及び曲線１６並びに曲線７及び曲線１７に基づいて複数投影図頂点算出手段５８は３次元における座標を決定し、またそれぞれの曲線の点列の座標を算出して曲線６及び曲線１６並びに曲線７及び曲線１７の２次元ＮＵＲＢ曲線を生成する。その後、必要なデータが３次元ＮＵＲＢ曲線変換手段６０に入力されて復元処理され、表示手段２０６に立体形状として表示させる（操作８）。この場合、プリミティブな楕円として認識される。
【００３１】
操作者により曲線１８、２０及び２１が入力される。曲線１８が入力されても、組み合わせ対象がないため、同一投影図組合せ検出手段５２は処理を行わない。また複数投影図組合せ手段５６も他の投影面上においても曲線１８との組合せを検出できないので処理を行わない。曲線２０が入力されると、同一投影図組合せ検出手段５２は曲線１８と２０を組合せとして検出する。検出された曲線の頂点に基づいて、同一投影図頂点算出手段５４は、透視変換行列Ｍⁱにより変換を行い、３次元上における曲線１８及び２０の頂点の座標を決定する。座標が決定されると、曲線１８及び２０上の点列をサンプリングする。必要なデータが３次元ＮＵＲＢ曲線変換手段６０に入力されて復元処理され、表示手段２０６に立体形状として表示させる。また曲線２１が入力されると、同一投影図組合せ検出手段５２は面対称と判断し、同様の処理を行い、表示手段２０６に立体形状として表示させる（操作９）。
【００３２】
図５は３次元ＮＵＲＢ曲線変換手段６０の構成を表すブロック図である。図において７０は初期値設定手段であり、同一投影図組合せ手段５２又は複数投影図組合せ手段５４により検出された２次元ＮＵＲＢ曲線のノットベクトルＵ及びそれぞれの制御点につける重みＷⁱ _jの初期値を設定し、２次元ＮＵＲＢ曲線が点列近傍を通過する際の点列通過パラメータｕⁱ _kを推定する。７２は２次元ＮＵＲＢ曲線算出手段であり、２次元ＮＵＲＢ曲線のノットベクトル、点列通過パラメータ及びそれぞれの制御点重みの初期値に基づいて、投影面上の点列に２次元ＮＵＲＢ曲線を当てはめるために、２次元ＮＵＲＢ曲線の制御点の座標を決定する。７４は３次元ＮＵＲＢ曲線算出手段であり、２次元ＮＵＲＢ曲線算出手段７２で算出された１又は複数投影面上の２次元ＮＵＲＢ曲線の制御点に基づいて、３次元座標上に３次元ＮＵＲＢ曲線を当てはめるために、３次元ＮＵＲＢ曲線の制御点の座標を決定する。７６は再投影手段であり、手書き誤差により２次元ＮＵＲＢ曲線の制御点と一致しなくなった３次元ＮＵＲＢ曲線の制御点を各投影面に再投影して、２次元ＮＵＲＢ曲線の制御点の座標を修正し、それに基づいて各制御点の重みも修正する。７８はパラメータ再推定手段であり、再投影手段７６で修正された２次元ＮＵＲＢ曲線の制御点の座標及び各制御点の重みに基づいて各投影面上での点列通過パラメータを再推定する。８０は収束判定手段であり、あらためて算出された２次元ＮＵＲＢ曲線と点列との距離の総和を算出し、全ての投影面において距離の総和が収束条件を満たしているかどうかを判断し、収束条件を満たしていると判断すると、その時に決定された３次元ＮＵＲＢ曲線を出力する。
【００３３】
図６は３次元ＮＵＲＢ曲線変換手段６０の動作手順を示す図である。また、図７は２次元ＮＵＲＢ曲線と入力点列、３次元ＮＵＲＢ曲線の関係を示す図である。図５、図６及び図７に基づいて３次元ＮＵＲＢ曲線変換手段６０のそれぞれの手段の動作について更に詳しく説明する。
【００３４】
３次元ＮＵＲＢ曲線をＰ（ｕ）、そのノットベクトルをＵ、投影面上ｉに投影された２次元ＮＵＲＢ曲線をＰⁱ（ｕ）とする。図３では投影面が２つあるのでＰ¹（ｕ）及びＰ²（ｕ）が存在する。
【００３５】
初期値設定手段７０では、ノットベクトルＵは２次元ＮＵＲＢ曲線の始点及び終点で多重度ｎ、間隔１（等間隔）のユニフォームに設定する。また各制御点重みＷⁱ _jを１．０とする。また点列通過パラメータをｕⁱ _kとすると、２次元ＮＵＲＢ曲線をＰⁱ（ｕ）が点列Ｓⁱ _kの近傍を通過する際の点列通過パラメータｕⁱ _kは次式（３）で推定される（図６ａ）。
【００３６】
【数２】

【００３７】
２次元ＮＵＲＢ曲線算出手段７２では、各制御点重みＷⁱ _j及び算出した点列通過パラメータをｕⁱ _kに基づいて各投影面の２次元ＮＵＲＢ曲線の制御点を求める。Ｐⁱ _jを２次元ＮＵＲＢ曲線の制御点、Ｎⁿ _j（ｕ）をノットベクトルＵ上のｎ次B-Spline関数とすると、次式（４）が成立する。
【００３８】
【数３】

【００３９】
また２次元ＮＵＲＢ曲線をＰⁱ（ｕ）が点列Ｓⁱ _kの近傍を通過する条件等により、次式（５）が成立する。
【００４０】
【数４】

【００４１】
式（６）に最小二乗法を適用し、決定された値を式（４）に代入することで、２次元ＮＵＲＢ曲線の制御点Ｐⁱ ₁、…、Ｐⁱ _N-1が求められる。なお、制御点Ｐⁱ ₁、…、Ｐⁱ _N-1は式（４）を満たす座標の組として求められ、最小二乗法により制御点の座標は一意に決まる（図６ｂ）。
【００４２】
３次元ＮＵＲＢ曲線算出手段７４は、２次元ＮＵＲＢ曲線算出手段７２で算出された１又は複数投影面上の２次元ＮＵＲＢ曲線の制御点に基づいて、３次元ＮＵＲＢ曲線の制御点の座標を決定する。ＮＵＲＢ曲線の性質より、３次元ＮＵＲＢ曲線の制御点と透視変換行列Ｍⁱ＝｛ｍⁱ _αβ｝によってある投影面に変換された２次元ＮＵＲＢ曲線の制御点とは１対１に対応する。したがって３次元ＮＵＲＢ曲線の制御点Ｐ_j（ｐ_jx，ｐ_jy，ｐ_jz）と２次元ＮＵＲＢ曲線の制御点Ｐⁱ _j（ｐⁱ _jx, ｐⁱ _jy,0）との間には次式（６）及び（７）の関係が成立する。
【００４３】
【数５】

【００４４】
ここで投影面がν面存在する場合、ｉ＝１，２，…，νについてそれぞれ式（６）及び（７）が成立するので、２ν本の連立方程式が算出される。これらの連立方程式を最小二乗法に基づいてｐ_jx，ｐ_jy，ｐ_jzについて解くことによって、それぞれの投影面上の同じ制御点に最も近い点を３次元ＮＵＲＢ曲線の制御点Ｐ_j（ｐ_jx，ｐ_jy，ｐ_jz）とすることができる（図６ｃ）。
【００４５】
再投影手段７６では３次元ＮＵＲＢ曲線の制御点Ｐ_j（ｐ_jx，ｐ_jy，ｐ_jz）を再投影する。Ｐ_j（ｐ_jx，ｐ_jy，ｐ_jz）は各投影面の２次元ＮＵＲＢ曲線の制御点Ｐⁱ _j（ｐⁱ _jx, ｐⁱ _jy,0）の最小二乗法により算出された座標値である。スケッチ誤差がない場合には、各投影面の制御点と３次元座標の制御点は一致し、３次元ＮＵＲＢ曲線算出手段７４により算出された３次元ＮＵＲＢ曲線の制御点Ｐ_jを再投影させなくてもよい。しかし、一般的には手書きの場合にはスケッチ誤差は生じるので、現在設定されている３次元ＮＵＲＢ曲線の制御点Ｐ_j（ｐ_jx，ｐ_jy，ｐ_jz）を２次元ＮＵＲＢ曲線の制御点Ｐⁱ _j（ｐⁱ _jx, ｐⁱ _jy,0）として各投影面に再投影しても同一座標にはならない。そこで、各投影面において、再投影した座標を新たな２次元ＮＵＲＢ曲線の制御点Ｐⁱ _j（ｐⁱ _jx, ｐⁱ _jy,0）と設定する。この新たな２次元ＮＵＲＢ曲線の制御点の座標を式（４）、（５）に再度入力し、初期値を１．０としている各制御点重みＷⁱ _jの修正を行う。なお、この修正した各制御点重みＷⁱ _jは式（４）、（５）を満たす組として求められ、最小二乗法により各制御点重みは一意に決まる（図６ｄ）。
【００４６】
パラメータ再推定手段７８は新たに算出された２次元ＮＵＲＢ曲線の制御点Ｐⁱ _j（ｐⁱ _jx, ｐⁱ _jy,0）及び各制御点重みＷⁱ _jに基づいて、再度点列通過パラメータｕⁱ _kの推定を行う。ＮＵＲＢ曲線の場合、投影された２次元上の座標点と投影した３次元上の座標点との曲線上のパラメータが同じになるため、投影された２次元ＮＵＲＢ曲線のパラメータが明示的に算出され、入力点列位置のパラメータを推定することができる。そこで、投影面上の点列Ｓⁱ _kとの最小距離を与える点をＰⁱ（ｕⁱ _k）とすると、投影面上のベクトルＳⁱ _k−Ｐⁱ（ｕⁱ _k）とＰⁱ（ｕⁱ _k）が直交するという条件が成立するから、次式（８）が成立する。
【００４７】
【数６】

【００４８】
したがってこれを満たすようなＰⁱ（ｕⁱ _k）を求めることにより、あらためてｕⁱ _kを決定する（図６ｅ）。
【００４９】
収束判定手段８０は、投影面上の点列Ｓⁱ _kと２次元ＮＵＲＢ曲線Ｐⁱ（ｕ）との総和を次式（９）に基づいて計算する。
【００５０】
【数７】

【００５１】
全ての投影面においてこのｅの値が前回のｅの値との変化率が０になるまで（収束したと判断するまで）繰り返す。すなわち、収束したと判断しなかったときは、算出した各制御点重みＷⁱ _j及び算出した点列通過パラメータｕⁱ _kを２次元ＮＵＲＢ曲線算出手段７２に再度入力し、収束したと判断されるまで７２〜８０の各手段の動作を繰り返し行う（図６ｆ）。
【００５２】
全ての投影面において変化率が０になり、収束したと判断した場合、このときの３次元ＮＵＲＢ曲線の制御点Ｐ_j（ｐ_jx，ｐ_jy，ｐ_jz）等の３次元ＮＵＲＢ曲線による立体形状のデータを入出力機構２０５を介して表示装置２０６に表示させる。また、記憶処理部２０３を介して図形記憶領域２０４にデータ保存させておくことも可能である。
【００５３】
以上のように第１の実施の形態によれば、スケッチ入力手段２０９を介して操作者により入力された投影面における手書き曲線に基づいて、幾何処理部は２次元ＮＵＲＢ曲線を作成し、１又は複数投影面において立体形状としての組合せとなる曲線の座標等を調整しながら３次元ＮＵＲＢ曲線に変換し、立体形状を復元して表示装置２０７に表示するようにしたので、ＮＵＲＢ曲線の特性を用いて制御点間の幾何関係のみから３次元ＮＵＲＢ曲線を算出することができるため、入力点列に含まれる手書きの誤差や微小振動が３次元形状に影響を及ぼさずに３次元自由曲線として自由度の高い３次元ＮＵＲＢ曲線を生成し、立体形状を表示することができる。また、ＮＵＲＢ曲線による表現が可能となったので、円や円弧のようなプリミティブな形状を生成することもできる。
【００５４】
実施形態２．
なお、上述の第１の実施の形態においては、入力曲線として手書き曲線を例示したが、点の集合としての曲線の座標が入力でき、投影面が特定されるような装置であれば実施は可能である。
【００５５】
【発明の効果】
以上のように本発明によれば、１又は複数の投影面を決定し、その投影面に投影された立体形状の２次元形状を２次元自由曲線として入力し、有理曲線である３次元ＮＵＲＢ曲線を算出し、立体形状を復元するので、有理多項式曲線であるＮＵＲＢ曲線の性質により、正確な立体形状を復元することが可能である。また円や円弧のようなプリミティブな形状の２次曲線に対しても同様に理論的に正確な２次曲線を復元することが可能である。さらに、２次元自由曲線から３次元ＮＵＲＢ曲線へ変換において、微小変化による影響を受けることが少ないので、手書きのような誤差を含む２次元自由曲線に対しても滑らかな３次元ＮＵＲＢ曲線への変換が可能になり、簡単な入力で操作者のイメージに近い立体形状が得られる。
【００５６】
また、本発明によれば、立体形状を表示する座標空間上に表示された補助用直方体を回転移動又は平行移動させることにより投影面を決定するので、視覚的に投影面を決定しやすくなる。
【００５７】
また、本発明によれば、同一投影面上において面対称関係にある２次元自由曲線又は複数投影面上において立体形状の同一線を表すものと見なされる２次元自由曲線の組み合わせを抽出し、それぞれの２次元自由曲線をアフィン変換してそれぞれの端点の座標の補正を行うので、１又は複数投影面上における立体形状の頂点を調整することができる。
【００５８】
また、本発明によれば、点列間の距離の比例関係に基づいて点列通過パラメータを推定し、点列通過パラメータに基づいて２次元ＮＵＲＢ曲線の制御点の座標を算出し、２次元ＮＵＲＢ曲線の制御点及び透視変換行列に基づいて３次元ＮＵＲＢ曲線の制御点の座標を算出し、３次元ＮＵＲＢ曲線を算出するので、表現できる曲線に制限がない有理多項式曲線であるＮＵＲＢ曲線の性質により、入力された曲線に対しても正確に表現することができる。
【００５９】
また、本発明によれば、２次元ＮＵＲＢ曲線の制御点を変数とする方程式を算出し、３次元における端点の座標、点列の座標及び点列のパラメータに基づいて、点列との誤差が最小となるような２次元ＮＵＲＢ曲線を算出して、方程式に基づいて２次元ＮＵＲＢ曲線の制御点の座標を決定するので、複数投影面における同一曲線の手書き誤差による曲率等の誤差を抑えられる。
【００６０】
また、本発明によれば、透視変換行列に基づいた２次元ＮＵＲＢ曲線の制御点と３次元ＮＵＲＢ曲線の制御点の座標との関係式を算出し、複数の投影面間で同一の制御点の関係を表す関係式において、誤差が最小になるように関係式を解き、３次元ＮＵＲＢ曲線の制御点の座標を算出するので、複数投影面における同一曲線の手書き誤差による曲率等の誤差を抑えられる。
【００６１】
また、本発明によれば、更に、算出された３次元ＮＵＲＢ曲線の制御点の座標を、透視変換行列に基づいて１又は複数の投影面に再投影し、再投影により得られる２次元ＮＵＲＢ曲線の制御点の座標を新たな２次元ＮＵＲＢ曲線の制御点の座標として点列のパラメータを再推定し、点列と算出された２次元ＮＵＲＢ曲線との距離の関係の変化率が収束したものと判断するまで２次元ＮＵＲＢ曲線の制御点の座標及び点列のパラメータを再推定し、３次元ＮＵＲＢ曲線を算出するので、複数投影面における同一曲線の手書き誤差による曲率等の誤差を抑えられ、実際に意図していた立体形状に近づけて表示させることができる。
【図面の簡単な説明】
【図１】本発明の第１の実施の形態に係る透視図３次元ＮＵＲＢ曲線自動復元方法を実施するためのブロック図である。
【図２】図１のブロック図を実際にハード構成する場合の構成図である。
【図３】幾何処理部２０２の構成を示すブロック図である。
【図４】自由度の高い曲線や円等のプリミティブな曲線を生成する入出力動作と幾何処理部２０２との関係を示す図である。
【図５】３次元ＮＵＲＢ曲線変換手段６０の構成を表すブロック図である。
【図６】３次元ＮＵＲＢ曲線変換手段６０の動作手順を示す図である。
【図７】２次元ＮＵＲＢ曲線と入力点列、３次元ＮＵＲＢ曲線の関係を示す図である。
【図８】投影面１を示す透視投影図９０である。
【図９】投影面２を示す透視投影図９２である。
【符号の説明】
２０１コマンド解析／処理部
２０２幾何処理部
２０３記憶処理部
２０４図形記憶領域
２０５入出力装置
２０６表示装置
２０７マウス
２０８キーボード
２０９スケッチ入力手段
５０投影面決定手段
５２同一投影図組合せ検出手段
５４同一投影図頂点算出手段
５６複数投影図組合せ検出手段
５８複数投影図頂点算出手段
６０３次元ＮＵＲＢ曲線変換手段
７０初期値設定手段
７２２次元ＮＵＲＢ曲線算出手段
７４３次元ＮＵＲＢ曲線算出手段
７６再投影手段
７８パラメータ再推定手段
８０収束判定手段[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a three-dimensional shape input method in a computer-aided three-dimensional input system, and in particular, from a two-dimensional perspective view to a three-dimensional in a CAID (computer-aided industrial design) / CAD (computer-aided design) / CG (computer graphics) system. The present invention relates to a three-dimensional automatic restoration system that restores a curve and displays it as a three-dimensional shape.
[0002]
[Prior art]
Although there are individual differences, an operator including a designer (that is, a human being) can combine a perspective drawing drawn in two dimensions and image a three-dimensional solid shape including a free curve in his / her head.
[0003]
8 and 9 are two-dimensional perspective projection images for imaging such a three-dimensional solid shape. FIG. 8 is a perspective projection diagram 90 showing the projection plane 1, and FIG. 9 is a perspective projection diagram 92 showing the projection plane 2.
[0004]
Previously, Journal of Japan Society for Precision Engineering, vol.59, no.6, 1993, Japan Society for Precision Engineering, Japan, Furushima, Kanai, Takahashi, “Generation of 3D free curve model by automatic recognition of handwritten figures” As described in Journal D, J71-D (1988) 84, IEICE, Japan, Toshimitsu Tanaka and others, "Reconstruction of 3D shape based on generalized symmetry" A three-dimensional automatic restoration system has been developed that restores a three-dimensional solid shape from such a two-dimensional perspective projection and supports an operator's three-dimensional image.
[0005]
In such a conventional three-dimensional automatic restoration system, a two-dimensional curve drawn on a perspective projection diagram is once represented by a two-dimensional point sequence, and a three-dimensional representation is made based on the point sequence of one or a plurality of perspective projection diagrams. The three-dimensional shape is restored by applying a three-dimensional curve to the three-dimensional point sequence.
[0006]
[Problems to be solved by the invention]
However, as described above, when a three-dimensional free curve is restored from one or a plurality of perspective projection views, the two-dimensional free curve is represented by a two-dimensional point sequence and two-dimensional points on one or a plurality of projection planes. Based on the sequence, it was converted into a 3D point sequence, and the 3D shape was restored by applying a 3D curve to the 3D sequence. The perspective projection view of can not be input or input is difficult. In addition, a quadratic curve such as a circle or arc on a three-dimensional surface cannot be expressed theoretically and accurately.
[0007]
The present invention has been made to solve the above-described problems, and a two-dimensional free curve containing a handwritten error or a quadratic curve expressing a circle or an arc is input. The purpose is to obtain a three-dimensional automatic restoration system that can be expressed as a smooth three-dimensional curve or a smooth quadratic curve on a three-dimensional surface, and can obtain a three-dimensional shape close to the operator's image with a simple input. To do. Here, the free curve includes straight lines and the like.
[0008]
[Means for Solving the Problems]
  The perspective view three-dimensional NURB curve automatic restoration method according to the present invention is:Projection plane determination meansDetermining one or more projection planes;Were determinedOn one or more projection planesAgainstA two-dimensional free curve that forms a two-dimensional shape when a three-dimensional shape is projected onto the projection surfaceIs entered,3D NURB curve conversion meansCalculate 3D NURB curve based on 2D free curveAnd one or more projection planesAnd a step of restoring a three-dimensional shape based on a three-dimensional NURB curve.
  In the present invention, one or a plurality of projection planes are determined, and a two-dimensional free curve constituting a two-dimensional shape when a three-dimensional shape is projected onto the projection plane is input by handwriting or the like. Based on the two-dimensional free curve, a three-dimensional NURB curve is calculated which has no limitation on the curve that can be drawn and is less affected by minute changes, and a three-dimensional shape is restored based on the three-dimensional NURB curve.
[0009]
  Further, one or more projection planes of the perspective view three-dimensional NURB curve automatic restoration method according to the present invention are:Displayed by the display means input via the input meansOn 3D coordinate spaceofAuxiliary rectangular parallelepipedAgainstRotation or translationBased on the instruction, the projection plane determining meansdecide.
  In the present invention, the auxiliary rectangular parallelepiped displayed on the three-dimensional coordinate space for restoring the three-dimensional shape is rotated or translated to visually display the auxiliary rectangular parallelepiped from the viewpoint corresponding to the projection plane. Determine the face.
[0010]
  Also, a perspective view three-dimensional NURB curve automatic restoration method according to the present invention3D NURB curve conversion meansThe step of calculating the three-dimensional NURB curve extracts a combination of two-dimensional free curves that are considered to represent two-dimensional free curves that are plane-symmetrical on the same projection plane or three-dimensional identical lines on a plurality of projection planes. , Affine transformation of each two-dimensional free curve to correct the coordinates of each end pointCalculated and estimatedThe coordinates of the control points of the two-dimensional NURB curve are calculated based on the value of the point sequence passage parameter when the two-dimensional NURB curve passes through the vicinity of each point sequence, and the control points and perspective transformation matrix of the two-dimensional NURB curve are calculated. A three-dimensional NURB curve is calculated based on the coordinates of the control points of the three-dimensional NURB curve calculated based on this.
  In the present invention, a two-dimensional free curve that is plane-symmetrical on the same projection plane or a combination of two-dimensional free curves that are considered to represent the same solid line on a plurality of projection planes is extracted. The free curve is calculated by correcting the coordinates of the end points by affine transformation. Further, the value of the point sequence passing parameter when the two-dimensional NURB curve passes in the vicinity of each point sequence is estimated, and the coordinates of the control point of the two-dimensional NURB curve are calculated based on the value of the point sequence passing parameter. A three-dimensional NURB curve is calculated based on the control point of the three-dimensional NURB curve and the coordinates of the control point of the three-dimensional NURB curve calculated based on the perspective transformation matrix.
[0011]
Further, the point sequence passage parameter of the perspective view three-dimensional NURB curve automatic restoration method according to the present invention is set based on the proportional relationship of the distance between the point sequences. In the present invention, the point sequence passing parameter when the two-dimensional NURB curve passes in the vicinity of each point sequence is set in proportion to the distance between the point sequences.
[0012]
Further, in the perspective view three-dimensional NURB curve automatic restoration method according to the present invention, the coordinates of the control points of the two-dimensional NURB curve are calculated using the control points of the two-dimensional NURB curve as variables. Based on the coordinates of the end points, the coordinates of the point sequence, and the value of the point sequence passing parameter, the value of the variable that minimizes the error from the point sequence is determined as the coordinate of the control point of the two-dimensional NURB curve.
In the present invention, an equation using the control point of the two-dimensional NURB curve as a variable is calculated. In the equation, the value of a variable that minimizes the error from the point sequence among the relational expressions between the control points obtained from the coordinates of the end points of the two-dimensional free curve, the coordinates of the point sequence and the point sequence passing parameters is 2 It is determined as the coordinate of the control point of the dimension NURB curve.
[0013]
In addition, the calculation of the coordinates of the control points of the three-dimensional NURB curve in the perspective view three-dimensional NURB curve automatic restoration method according to the present invention is performed using the control points of the two-dimensional NURB curve and the control points of the three-dimensional NURB curve based on the perspective transformation matrix. The relational expression with respect to the coordinates of the three-dimensional coordinates is calculated, and in the relational expression representing the relation between the same control points among a plurality of projection planes, the relational expression is solved with respect to the three-dimensional coordinates so as to minimize the error, and the control of the three-dimensional NURB curve Calculate the coordinates of the point.
In the present invention, a relational expression between the control point of the two-dimensional NURB curve and the coordinate of the control point of the three-dimensional NURB curve based on the perspective transformation matrix is calculated. In a relational expression representing the relationship between the same control points between a plurality of projection planes, the relational expression for the three-dimensional coordinates is solved so that the error is minimized, and the three-dimensional that minimizes the handwriting error between the plurality of projection planes. The coordinates of the control points of the NURB curve are calculated.
[0014]
  Also, a perspective view three-dimensional NURB curve automatic restoration method according to the present invention3D NURB curve conversion meansThe step of calculating the three-dimensional NURB curve further re-projects the coordinates of the calculated control points of the three-dimensional NURB curve on one or a plurality of projection planes based on the perspective transformation matrix, and the two-dimensional NURB obtained by the reprojection. Using the coordinates of the control points of the curve as the coordinates of the control points of the new two-dimensional NURB curve, the point sequence passing parameters are re-estimated, and the rate of change in the distance relationship between the point sequence and the calculated two-dimensional NURB curve has converged The coordinates of the control points of the two-dimensional NURB curve and the parameters of the point sequence are re-estimated until it is determined that the three-dimensional NURB curve is obtained.
  In the present invention, the calculated coordinates of the control points of the three-dimensional NURB curve are reprojected on one or a plurality of projection planes. Convert the control points of the 3D NURB curve with minimal errors to the control points of the 2D NURB curve, re-estimate the point sequence passing parameters as control points of the new 2D NURB curve, Based on this, the change rate of the relationship between the distance between the point sequence and the calculated two-dimensional NURB curve converges, and the coordinates of the control points of the two-dimensional NURB curve and the parameters of the point sequence are re-estimated until the error is suppressed. A dimensional NURB curve is calculated.
[0015]
DETAILED DESCRIPTION OF THE INVENTION
FIG. 1 is a block diagram for carrying out the perspective three-dimensional NURB curve automatic restoration method according to the first embodiment of the present invention. In the figure, reference numeral 201 denotes a command analysis / processing unit, which is connected to a display device 206 and sketch input means 209 via an input / output mechanism 205, and analyzes and processes input commands. A geometric processing unit 202 restores a three-dimensional shape based on the coordinate values of a two-dimensional free curve given as a set of coordinate values of points. A storage processing unit 203 performs processing for storing data in the graphic storage area 204. Reference numeral 205 denotes an input / output mechanism, which is an interface between the command analysis / processing unit 201, the geometric processing unit 202, the storage processing unit 203, and the graphic information storage area 204, and the display device 206 and the sketch input unit 209. It is. The sketch input unit 209 is a part to which an operator's operation is input, and includes a mouse 207 and a keyboard 208.
[0016]
FIG. 2 is a block diagram when the block diagram of FIG. 1 is actually configured in hardware. The perspective view three-dimensional NURB curve automatic restoration method is mainly restored on a computer. In FIG. 2, a central computer 101, a main storage device 102, an input / output control device 103, a mouse or tablet 104, a keyboard 105, an external storage device 106, and a display device 107 are connected by a system bus 108 (or a similar mechanism). The system 100 is configured. In some cases, the computer system is similar to this.
[0017]
FIG. 3 is a block diagram showing a configuration of the geometric processing unit 202. In the figure, reference numeral 50 denotes a projection plane determining means 50, which displays a rectangular parallelepiped that assists in determining the projection plane on the display device 206, and changes the viewpoint of the rectangular parallelepiped according to a rotation or movement instruction from the sketch input means. When the operator determines the viewpoint, the perspective transformation matrix of the viewpoint is calculated and output. The same projection map combination detection means 52 detects a combination of a vertex (curve start point and end point) and a two-dimensional curve in a plane symmetry relationship in a two-dimensional handwritten curve of a single perspective projection map. 54 is the same projection view vertex calculation means, which calculates the three-dimensional coordinate position of the vertex detected by the same projection view combination means 52. Reference numeral 56 denotes a plurality of projection map combination means for detecting a combination of a vertex (start point and end point of a curve) and a two-dimensional free curve associated with the same vertex / curve in space in a two-dimensional handwritten curve of a plurality of perspective projection diagrams. To do. Reference numeral 58 denotes a plurality of projection map vertex calculation means for calculating the three-dimensional coordinate position of the vertex detected by the plurality of projection map combination means 56. Reference numeral 60 denotes a three-dimensional NURB curve conversion means, which is a two-dimensional NURB curve detected by the same projection view combination detection means 52 or the plurality of projection view combination detection means 54, the same projection view vertex calculation means 56, or a plurality of projection view vertex calculation means 58. Based on the vertex coordinates calculated in, the input point sequence coordinates on the projection plane, and the perspective transformation matrix of the projection plane, vertices and curves are corrected by performing affine transformation (scaling, deformation in the x-axis or y-axis direction, etc.) The three-dimensional NURB curve is restored from the two-dimensional free curve.
[0018]
Here, the NURB curve will be described. The NURB curve is a NonUniform Rati-onal B-Spline curve. When a curve on a plane is expressed parametrically using the parameter t, for example, a parameter curve based on a cubic polynomial is expressed by the following equation (1).
R (t) = t^ThreeP_Three+ T²P₂+ TP₁+ P₀                  ... (1)
Where P_Three, P₂, P₁And P₀Is a vector. Bezier curves and B-Spline curves are expressed in this parametric curve format. Bezier curves are determined by points called control points, and B-Spline curves must be connected independently of control points. Can do.
[0019]
These parametric curves are limited in the shape that can be accurately expressed, and one of the methods for accurately expressing them is an expression expressed in the form of a rational polynomial called a rational parametric curve. The NURB curve is one of rational parametric curves and is represented by the following equation (2).
[0020]
[Expression 1]

[0021]
Where w_iIs the control point P_iIs the weight corresponding to N_{i, p}(T) is the same as the B-Spline function. p is the order of the curve segment. This is a NURB curve.
[0022]
FIG. 4 is a diagram showing a relationship between an input / output operation for generating a primitive curve such as a curve or a circle with a high degree of freedom and the geometric processing unit 202. This figure is based on the perspective projections of FIGS. The geometric processing unit 202 will be described in detail with reference to FIGS. 3, 4, 8, and 9.
[0023]
The operator designates the projection plane 1 as a viewpoint in the drawing direction of the two-dimensional curve from the sketch input means 209 (operation 1). The projection plane 1 includes a two-dimensional free curve that is considered to be plane symmetric. A projection plane including such a curve is selected first. In order to specify the projection plane, an auxiliary rectangular parallelepiped is displayed in advance on the display means 206 by the projection plane determining means 50, and the operator rotates or moves the rectangular parallelepiped via the sketch input means 209. Let me decide.
[0024]
When the operator designates the projection plane 1 as a viewpoint in the drawing direction, the operator performs handwriting input of a two-dimensional curve. Even if the curve 1 is input, since there is no combination target, the same projection map combination detection unit 52 does not perform processing. Further, since the multiple projection map combination means 56 has only one projection surface, no processing is performed at this time. When the curves 2, 3 and 4 are input, the same projection map combination detection means 52 detects that the combination target with the curve 1 is input. Based on the detected end points of the curve (the vertices of the three-dimensional shape), the same projection view vertex calculation means 54 performs the perspective transformation matrix M corresponding to the projection plane determined by the projection plane determination means 50.ⁱTo convert the vertices of the curves 1 to 4 on the three dimensions. When the coordinates are determined, the point sequence on the curves 1 to 4 is sampled. Necessary data is input to the three-dimensional NURB curve converting means 60 and restored, and displayed on the display means 206 as a three-dimensional shape (operation 2). Although it is a curve with a high degree of freedom that includes a handwriting error and the curvature changes, it is not affected by a minute change due to the nature of the NURB curve, so that a smooth three-dimensional NURB curve can be generated.
[0025]
Next, the operator inputs the curve 5. It is assumed that the same projection map combination detection unit 52 does not have a plane symmetric curve on the projection diagram 90 with respect to the curve 5. The three-dimensional shape displayed on the display means 206 remains as it is (operation 3). When the same curve from another projection plane is recognized by the plural projection map combination detection means 56, it is three-dimensionalized.
[0026]
In operation 4, the curve 6 and the curve 7 are input as ellipses by the input using the two-dimensional NURB curve. Since there are no corresponding curves on the projection plane 1 for the curves 6 and 7, the three-dimensional shape displayed on the display means 206 remains as it is (operation 4). The curves 6 and 7 are also three-dimensionalized when the same projection from different projection planes is recognized by the multiple projection combination detection means 56.
[0027]
When the curve 8 is input, the same projection map combination detection unit 52 assumes that no curve corresponding to the curve 8 exists on the projection map 90. The same projection view vertex calculation means 54 determines that the curve 9 itself exists as a plane-symmetric curve, and the perspective projection vertex calculation means 54ⁱTo determine the coordinates of the vertex of the curve 9 in three dimensions, and sample the curved point sequence. Necessary data is input to the three-dimensional NURB curve converting means 60, and the control points of the three-dimensional NURB curve are output. The three-dimensional shape display control means 62 displays a three-dimensional shape on the display means 64 based on the output control points of the three-dimensional NURB curve (operation 5).
[0028]
When the input of the two-dimensional curve by handwriting from the projection plane 1 is completed, the operator again rotates and moves the auxiliary rectangular parallelepiped displayed on the display device 206 via the sketch input unit 209, and the projection plane 2. Is designated as the viewpoint in the drawing direction of the two-dimensional curve (operation 6). Here, the curves 11, 12, 19 corresponding to the curves 1, 2, 3, 4 and 9 whose positions in the three-dimensional shape have been determined are already on the projection plane 2 with respect to the viewpoints of these positions. It is drawn. Therefore, another curve is input based on the curve.
[0029]
When the curve 15 is input by the operator, since there is no combination target on the same projection plane, the same projection map combination detection unit 52 does not perform processing. Based on the projection planes 1 and 2, the multiple-projection drawing combination unit 56 determines that the curve 5 and the curve 15 represent the same curve. Based on the curves 5 and 15, the multiple projection figure vertex calculation means 58 determines the coordinates of the vertices, calculates the coordinates of the point sequence of each curve, and generates the two-dimensional NURB curves of the curves 5 and 15. Thereafter, necessary data is input to the three-dimensional NURB curve converting means 60 and restored, and displayed on the display means 206 as a three-dimensional shape (operation 7).
[0030]
When the curves 16 and 17 are input by the operator, there is no combination target on the same projection plane, so the same projection combination detection unit 52 does not perform processing. Based on the projection planes 1 and 2, the multiple projection map combination unit 56 determines that the curves 6 and 16 and the curves 7 and 17 represent the same curve. Based on the curves 6 and 16, and the curves 7 and 17, the multiple projection figure vertex calculation means 58 determines the coordinates in three dimensions, and calculates the coordinates of the point sequence of each curve to obtain the curves 6, 16 and 16 7 and curve 17 are generated as a two-dimensional NURB curve. Thereafter, necessary data is input to the three-dimensional NURB curve converting means 60 and restored, and displayed on the display means 206 as a three-dimensional shape (operation 8). In this case, it is recognized as a primitive ellipse.
[0031]
Curves 18, 20 and 21 are input by the operator. Even if the curve 18 is input, since there is no combination target, the same projection map combination detection unit 52 does not perform processing. Further, the multiple projection combination means 56 does not perform processing because the combination with the curve 18 cannot be detected on other projection planes. When the curve 20 is input, the same projection combination detection means 52 detects the curves 18 and 20 as a combination. On the basis of the detected vertex of the curve, the same projection view vertex calculation means 54 calculates the perspective transformation matrix M.ⁱIs converted to determine the coordinates of the vertices of the curves 18 and 20 in three dimensions. Once the coordinates are determined, the point sequence on curves 18 and 20 is sampled. Necessary data is input to the three-dimensional NURB curve converting means 60 and restored, and displayed on the display means 206 as a three-dimensional shape. When the curve 21 is input, the same projection combination detection unit 52 determines that the plane is symmetrical, performs the same processing, and causes the display unit 206 to display the solid shape (operation 9).
[0032]
FIG. 5 is a block diagram showing the configuration of the three-dimensional NURB curve converting means 60. In the figure, reference numeral 70 denotes an initial value setting means, which is a knot vector U of a two-dimensional NURB curve detected by the same projection map combination means 52 or the plurality of projection map combination means 54 and a weight W applied to each control point.ⁱ _jThe initial value of is set, and the point sequence passing parameter u when the two-dimensional NURB curve passes near the point sequenceⁱ _kIs estimated. 72 is a two-dimensional NURB curve calculation means for applying a two-dimensional NURB curve to a point sequence on the projection plane based on a knot vector of the two-dimensional NURB curve, a point sequence passing parameter, and initial values of respective control point weights. Next, the coordinates of the control point of the two-dimensional NURB curve are determined. 74 is a three-dimensional NURB curve calculating means. Based on the control points of the two-dimensional NURB curve on one or a plurality of projection planes calculated by the two-dimensional NURB curve calculating means 72, a three-dimensional NURB curve is displayed on the three-dimensional coordinates. For fitting, the coordinates of the control points of the three-dimensional NURB curve are determined. 76 is reprojection means, which reprojects the control points of the three-dimensional NURB curve, which are no longer coincident with the control points of the two-dimensional NURB curve, due to handwriting errors, to obtain the coordinates of the control points of the two-dimensional NURB curve. The weight of each control point is also corrected based on the correction. Reference numeral 78 denotes parameter re-estimation means, which re-estimates the point sequence passing parameters on each projection plane based on the coordinates of the control points of the two-dimensional NURB curve corrected by the re-projection means 76 and the weights of the respective control points. Reference numeral 80 denotes convergence determination means, which calculates the sum of the distances between the newly calculated two-dimensional NURB curve and the point sequence, determines whether the sum of the distances satisfies the convergence condition on all projection planes, and the convergence condition If it is determined that the above is satisfied, the three-dimensional NURB curve determined at that time is output.
[0033]
FIG. 6 is a diagram showing an operation procedure of the three-dimensional NURB curve converting means 60. FIG. 7 is a diagram showing the relationship between a two-dimensional NURB curve, an input point sequence, and a three-dimensional NURB curve. The operation of each unit of the three-dimensional NURB curve converting unit 60 will be described in more detail with reference to FIGS.
[0034]
A three-dimensional NURB curve is P (u), its knot vector is U, and a two-dimensional NURB curve projected onto i on the projection plane is Pⁱ(U). In FIG. 3, there are two projection planes.¹(U) and P²(U) exists.
[0035]
In the initial value setting means 70, the knot vector U is set to a uniform with a multiplicity of n and an interval of 1 (equal interval) at the start and end points of the two-dimensional NURB curve. Each control point weight Wⁱ _jIs 1.0. Also, the point sequence passing parameter is uⁱ _kThen the two-dimensional NURB curve is Pⁱ(U) is the point sequence Sⁱ _kParameter passing parameter u when passing in the vicinity ofⁱ _kIs estimated by the following equation (3) (FIG. 6a).
[0036]
[Expression 2]

[0037]
In the two-dimensional NURB curve calculation means 72, each control point weight Wⁱ _jAnd the calculated point sequence passing parameter uⁱ _kBased on the above, the control points of the two-dimensional NURB curve of each projection plane are obtained. Pⁱ _jIs the control point of the two-dimensional NURB curve, Nⁿ _jWhen (u) is an n-order B-Spline function on the knot vector U, the following expression (4) is established.
[0038]
[Equation 3]

[0039]
Also, the two-dimensional NURB curve is Pⁱ(U) is the point sequence Sⁱ _kThe following equation (5) is established depending on the condition of passing through the vicinity of.
[0040]
[Expression 4]

[0041]
By applying the least square method to equation (6) and substituting the determined value into equation (4), control point P of the two-dimensional NURB curveⁱ ₁... Pⁱ _N-1Is required. Note that the control point Pⁱ ₁... Pⁱ _N-1Is obtained as a set of coordinates satisfying Equation (4), and the coordinates of the control point are uniquely determined by the least square method (FIG. 6b).
[0042]
The three-dimensional NURB curve calculation means 74 determines the coordinates of the control points of the three-dimensional NURB curve based on the control points of the two-dimensional NURB curve on one or a plurality of projection planes calculated by the two-dimensional NURB curve calculation means 72. . From the nature of the NURB curve, the control points of the three-dimensional NURB curve and the perspective transformation matrix Mⁱ= {Mⁱ _αβ} Has a one-to-one correspondence with the control points of the two-dimensional NURB curve converted into a certain projection plane. Therefore, the control point P of the three-dimensional NURB curve_j(P_jx, P_jy, P_jz) And two-dimensional NURB curve control point Pⁱ _j(Pⁱ _jx, pⁱ _jy, 0) holds the relationship of the following equations (6) and (7).
[0043]
[Equation 5]

[0044]
Here, when the projection plane is a ν plane, equations (6) and (7) are established for i = 1, 2,..., Ν, respectively, and 2ν simultaneous equations are calculated. These simultaneous equations are converted to p based on the least squares method._jx, P_jy, P_jz, The closest point to the same control point on each projection plane is the control point P of the three-dimensional NURB curve._j(P_jx, P_jy, P_jz(Fig. 6c).
[0045]
In the reprojection means 76, the control point P of the three-dimensional NURB curve_j(P_jx, P_jy, P_jz) Is reprojected. P_j(P_jx, P_jy, P_jz) Is the control point P of the two-dimensional NURB curve on each projection plane.ⁱ _j(Pⁱ _jx, pⁱ _jy, 0) is a coordinate value calculated by the least square method. When there is no sketch error, the control point of each projection plane coincides with the control point of the three-dimensional coordinate, and the control point P of the three-dimensional NURB curve calculated by the three-dimensional NURB curve calculation means 74 is obtained._jNeed not be reprojected. However, since a sketch error generally occurs in the case of handwriting, the control point P of the currently set three-dimensional NURB curve_j(P_jx, P_jy, P_jz) Is the control point P of the two-dimensional NURB curveⁱ _j(Pⁱ _jx, pⁱ _jy, 0), the same coordinates are not obtained even if reprojected onto each projection plane. Therefore, on each projection plane, the re-projected coordinates are used as control points P of a new two-dimensional NURB curve.ⁱ _j(Pⁱ _jx, pⁱ _jy, 0). The coordinates of the control points of this new two-dimensional NURB curve are input again into equations (4) and (5), and each control point weight W is set to an initial value of 1.0.ⁱ _jMake corrections. Each corrected control point weight Wⁱ _jIs obtained as a set satisfying equations (4) and (5), and each control point weight is uniquely determined by the least square method (FIG. 6d).
[0046]
The parameter re-estimation means 78 uses the newly calculated control point P of the two-dimensional NURB curve.ⁱ _j(Pⁱ _jx, pⁱ _jy, 0) and each control point weight Wⁱ _jAgain, the point sequence passing parameter uⁱ _kEstimate In the case of a NURB curve, the parameters of the projected two-dimensional coordinate point and the projected three-dimensional coordinate point are the same, so the parameters of the projected two-dimensional NURB curve are explicitly calculated. The parameter of the input point sequence position can be estimated. Therefore, the point sequence S on the projection planeⁱ _kThe point giving the minimum distance to Pⁱ(Uⁱ _k), The vector S on the projection planeⁱ _k-Pⁱ(Uⁱ _k) And Pⁱ(Uⁱ _k) Are orthogonal, the following equation (8) is satisfied.
[0047]
[Formula 6]

[0048]
Therefore P that satisfies thisⁱ(Uⁱ _k), U againⁱ _kIs determined (FIG. 6e).
[0049]
Convergence determining means 80 includes a point sequence S on the projection plane.ⁱ _kAnd two-dimensional NURB curve PⁱThe sum total with (u) is calculated based on the following equation (9).
[0050]
[Expression 7]

[0051]
This is repeated until the rate of change of the value of e from the previous value of e becomes zero (until it is determined that it has converged) on all projection planes. That is, when it is not determined that it has converged, each calculated control point weight Wⁱ _jAnd the calculated point sequence passing parameter uⁱ _kIs again input to the two-dimensional NURB curve calculating means 72, and the operations of the means 72 to 80 are repeated until it is determined that the two converged (FIG. 6f).
[0052]
When it is determined that the rate of change has become zero on all projection planes and has converged, the control point P of the three-dimensional NURB curve at this time_j(P_jx, P_jy, P_jz) And the like are displayed on the display device 206 via the input / output mechanism 205. It is also possible to store data in the graphic storage area 204 via the storage processing unit 203.
[0053]
As described above, according to the first embodiment, the geometric processing unit creates a two-dimensional NURB curve based on the handwritten curve on the projection plane input by the operator via the sketch input unit 209, Since the three-dimensional NURB curve is converted while adjusting the coordinates or the like of the curve as a combination as a three-dimensional shape on a plurality of projection planes, and the three-dimensional shape is restored and displayed on the display device 207, the characteristics of the NURB curve are used. Since the 3D NURB curve can be calculated only from the geometric relationship between the control points, the degree of freedom as a 3D free curve can be obtained without any handwritten error or minute vibration included in the input point sequence affecting the 3D shape. A three-dimensional NURB curve having a high height can be generated and a three-dimensional shape can be displayed. In addition, since it is possible to express with a NURB curve, a primitive shape such as a circle or an arc can be generated.
[0054]
Embodiment 2. FIG.
In the first embodiment described above, the handwritten curve is exemplified as the input curve, but the present invention can be implemented as long as the apparatus can input the coordinates of the curve as a set of points and specify the projection plane. It is.
[0055]
【The invention's effect】
As described above, according to the present invention, one or a plurality of projection planes are determined, a two-dimensional shape of a solid shape projected on the projection plane is input as a two-dimensional free curve, and a three-dimensional NURB curve that is a rational curve. Since the three-dimensional shape is restored, the accurate three-dimensional shape can be restored due to the nature of the NURB curve, which is a rational polynomial curve. Similarly, a theoretically accurate quadratic curve can be restored for a quadratic curve having a primitive shape such as a circle or an arc. Furthermore, since conversion from a two-dimensional free curve to a three-dimensional NURB curve is less affected by small changes, conversion to a smooth three-dimensional NURB curve is also possible for a two-dimensional free curve containing errors such as handwriting. Therefore, a three-dimensional shape close to the operator's image can be obtained with a simple input.
[0056]
Further, according to the present invention, the projection plane is determined by rotating or translating the auxiliary rectangular parallelepiped displayed on the coordinate space for displaying the three-dimensional shape, so that the projection plane can be easily determined visually.
[0057]
In addition, according to the present invention, a two-dimensional free curve that is plane-symmetrical on the same projection plane or a combination of two-dimensional free curves that are regarded as representing the same solid line on a plurality of projection planes is extracted, Since the two-dimensional free-form curve is affine transformed to correct the coordinates of the respective end points, the vertex of the three-dimensional shape on one or a plurality of projection planes can be adjusted.
[0058]
Further, according to the present invention, the point sequence passing parameter is estimated based on the proportional relationship of the distance between the point sequences, the control point coordinates of the two-dimensional NURB curve are calculated based on the point sequence passing parameter, and the two-dimensional NURB is calculated. Based on the control points of the curve and the perspective transformation matrix, the coordinates of the control points of the three-dimensional NURB curve are calculated, and the three-dimensional NURB curve is calculated. The input curve can be expressed accurately.
[0059]
In addition, according to the present invention, an equation having a control point of a two-dimensional NURB curve as a variable is calculated, and an error from the point sequence is calculated based on the coordinates of the end points, the coordinates of the point sequence, and the parameters of the point sequence in three dimensions. Since the minimum two-dimensional NURB curve is calculated and the coordinates of the control points of the two-dimensional NURB curve are determined based on the equation, errors such as curvature due to handwriting errors of the same curve on a plurality of projection planes can be suppressed.
[0060]
In addition, according to the present invention, a relational expression between the control point of the two-dimensional NURB curve and the coordinate of the control point of the three-dimensional NURB curve based on the perspective transformation matrix is calculated, and the same control point is calculated among a plurality of projection planes. In the relational expression representing the relation, the relational expression is solved so as to minimize the error, and the coordinates of the control points of the three-dimensional NURB curve are calculated, so that errors such as curvature due to handwriting errors of the same curve on a plurality of projection planes can be suppressed. .
[0061]
According to the present invention, the coordinates of the calculated control points of the three-dimensional NURB curve are further reprojected on one or a plurality of projection planes based on the perspective transformation matrix, and the two-dimensional NURB curve obtained by the reprojection. The coordinates of the point sequence are re-estimated using the control point coordinates of the new two-dimensional NURB curve as the control point coordinates, and the rate of change in the distance relationship between the point sequence and the calculated two-dimensional NURB curve is converged. Until the judgment is made, the coordinates of the control points of the two-dimensional NURB curve and the parameters of the point sequence are re-estimated and the three-dimensional NURB curve is calculated. Can be displayed close to the intended three-dimensional shape.
[Brief description of the drawings]
FIG. 1 is a block diagram for carrying out a perspective three-dimensional NURB curve automatic restoration method according to a first embodiment of the present invention.
FIG. 2 is a configuration diagram when the block diagram of FIG. 1 is actually configured in hardware.
3 is a block diagram showing a configuration of a geometric processing unit 202. FIG.
4 is a diagram illustrating a relationship between an input / output operation for generating a primitive curve such as a curve or a circle with a high degree of freedom and a geometric processing unit 202. FIG.
5 is a block diagram showing a configuration of a three-dimensional NURB curve conversion means 60. FIG.
6 is a diagram showing an operation procedure of a three-dimensional NURB curve converting means 60. FIG.
FIG. 7 is a diagram showing a relationship between a two-dimensional NURB curve and an input point sequence and a three-dimensional NURB curve.
FIG. 8 is a perspective projection view 90 showing the projection plane 1;
9 is a perspective projection view 92 showing the projection plane 2. FIG.
[Explanation of symbols]
201 Command analysis / processing section
202 Geometric processing unit
203 Storage processing unit
204 Figure storage area
205 I / O device
206 Display device
207 mouse
208 keyboard
209 Sketch input means
50 Projection plane determining means
52 Same projection view combination detection means
54 Same-projection vertex calculation means
56 Plural projection combination detection means
58 Multiple projection figure vertex calculation means
60 Three-dimensional NURB curve conversion means
70 Initial value setting means
72 Two-dimensional NURB curve calculation means
74 Three-dimensional NURB curve calculation means
76 Reprojection means
78 Parameter re-estimation means
80 Convergence judging means

Claims

A step projection plane determining means for determining one or more projection surfaces, determined against the one or more projection surfaces, a two-dimensional free curve constituting the two-dimensional shape when the projection of the three-dimensional shape to the projection surface Is input , the three-dimensional NURB curve conversion means calculates a three-dimensional NURB curve based on the two-dimensional free curve, and restores the three-dimensional shape based on the three-dimensional NURB curve of the one or more projection planes. A method of automatically restoring a perspective three-dimensional NURB curve.

A step projection plane determining means for determining one or more projection surfaces, wherein one or more of the projection plane, two-dimensional free curve input means constituting the two-dimensional shape when projected a three-dimensional shape to the projection surface It is input through, and based on the two-dimensional free curve input to the one or more projection surfaces,
A step of calculating the coordinates of the end points of the two-dimensional free curve, which is the vertex of the three-dimensional shape, and sampling the points on the two-dimensional free curve and calculating the coordinates of the point sequence When,
A step of the same and a plurality projection drawing vertex calculation means calculates the coordinates of the vertices of the three-dimensional shape in a coordinate and three-dimensional coordinates based on the perspective transformation matrix determined based on the projection plane of the end point of the two-dimensional free curve ,
A three- dimensional NURB curve converting means calculates a three-dimensional NURB curve based on the vertex on the three-dimensional coordinate, the coordinates of the point sequence, and the perspective transformation matrix, and the calculated three-dimensional NURB curve of the one or more projection planes. And a step of restoring a three-dimensional shape based on the three-dimensional NURB curve automatic restoration method.

3. The perspective view three-dimensional NURB curve automatic restoration method according to claim 1, further comprising a step of displaying the restored three-dimensional shape on a display means .

The one or more projection surfaces, wherein the input via the input means, based on the rotational movement or parallel movement instruction for auxiliary parallelepiped on the three-dimensional coordinate space is display means for displaying the projection plane determining means The method for automatically restoring a perspective three-dimensional NURB curve according to claim 1 or 2, wherein:

The step of calculating the three-dimensional NURB curve by the three-dimensional NURB curve converting means is considered to represent a two-dimensional free curve having a plane symmetry relationship on the same projection plane or a solid line having the same shape on a plurality of projection planes. A combination of two-dimensional free curves is extracted, each two-dimensional free curve is affine transformed and the coordinates of each end point are corrected and calculated, and the two-dimensional NURB curve passes through the vicinity of each estimated point sequence. The coordinates of the control points of the two-dimensional NURB curve are calculated based on the value of the point sequence passing parameter, and the control points of the three-dimensional NURB curve calculated based on the control points of the two-dimensional NURB curve and the perspective transformation matrix 3. The perspective view three-dimensional NURB curve automatic restoration method according to claim 2, wherein the three-dimensional NURB curve is calculated on the basis of the coordinates.

6. The perspective view three-dimensional NURB curve automatic restoration method according to claim 5, wherein the point sequence passage parameter is set based on a proportional relationship of distances between the point sequences.

The coordinates of the control points of the two-dimensional NURB curve are calculated using an equation having the control points of the two-dimensional NURB curve as variables, and the equations, the coordinates of the end points of the two-dimensional free curve, the coordinates of the point sequence, and the points 6. The perspective view according to claim 5, wherein a value of a variable that minimizes an error from the point sequence is determined as a coordinate of a control point of the two-dimensional NURB curve based on a value of a sequence passing parameter. Three-dimensional NURB curve automatic restoration method.

The calculation of the coordinate of the control point of the three-dimensional NURB curve is performed by calculating a relational expression between the control point of the two-dimensional NURB curve and the coordinate of the control point of the three-dimensional NURB curve based on the perspective transformation matrix. In the relational expression representing the relationship between the same control points between the faces, the relational expression is solved for the three-dimensional coordinates so that the error is minimized, and the coordinates of the control points of the three-dimensional NURB curve are calculated. The perspective view three-dimensional NURB curve automatic restoration method according to claim 5.

The step of calculating the three-dimensional NURB curve by the three-dimensional NURB curve conversion means further includes, based on the perspective conversion matrix, coordinates of the calculated control points of the three-dimensional NURB curve on the one or more projection planes. Re-projection, and using the coordinates of the control point of the two-dimensional NURB curve obtained by the re-projection as the coordinates of the control point of the new two-dimensional NURB curve, the value of the point sequence passing parameter is re-estimated and calculated as the point sequence Re-estimating the coordinates of the control points of the two-dimensional NURB curve and the point sequence passing parameter until it is determined that the rate of change in the distance relationship with the two-dimensional NURB curve has converged, and calculating the three-dimensional NURB curve The perspective view three-dimensional NURB curve automatic restoration method according to claim 5.

An input means for inputting instructions or coordinate data of a two-dimensional free curve;
Display means for displaying a two-dimensional or three-dimensional curve based on the coordinate data;
Projection plane determination for displaying an auxiliary rectangular parallelepiped on the display means and outputting a perspective transformation matrix based on one or more projection planes determined by a rotational movement instruction or a parallel movement instruction of the auxiliary rectangular parallelepiped input from the input means Means,
Same-projection-combination detection means for detecting a combination of two-dimensional free curves that are plane-symmetrical on the same projection plane based on the input coordinate data of the two-dimensional free curves and a point sequence of the two-dimensional free curves;
The same projection vertex calculation means for calculating the coordinates of the end points of the three-dimensional NURB curve based on the perspective transformation matrix and the vertex of the two-dimensional free curve that is plane-symmetric;
A plurality of projection map combination detection means for detecting a combination of the same curve on a three-dimensional shape and a point sequence of the two-dimensional free curve on a plurality of projection planes based on the input coordinate data of the two-dimensional free curve;
A plurality of projection drawing vertex calculation means for calculating the coordinates of the vertex of the three-dimensional NURB curve based on the perspective transformation matrix and the vertex of the same curve on the three-dimensional shape;
3D NURB curve conversion means for outputting coordinate data of the 3D NURB curve to the display means based on the perspective transformation matrix, the point sequence of the 2D free curve and the coordinates of the vertex of the 3D NURB curve. A perspective view three-dimensional NURB curve automatic restoration device characterized by