JPH0630996B2 - Quadratic curve generator - Google Patents

Description

The present invention relates to a quadratic curve creating tool suitable for use in teaching materials of geometry and polishing and forming quadric surfaces.

Conventionally, some instruments that draw and form quadratic curves use a link mechanism or many gears, but these are complicated in structure and difficult to manufacture and difficult to operate. It had a drawback.

INDUSTRIAL APPLICABILITY The present invention utilizes a common principle of a quadratic curve, which is different from the conventional one, and is easy to manufacture, and an accurate quadratic curve (parabola, ellipse, hyperbola) can be drawn by a simple operation. Its main purpose is to propose a quadratic curve generator.

First, the principle of the present invention will be described. Parabolas, ellipses and hyperbolas have the following common properties. In the parabola, as shown in Fig. 1, if the foot of the line below the quasi-line B from an arbitrary point P on the parabola A is Rp and the focus of the parabola A is F, the line segment PR
There is a first expression relationship between p and the line segment PF.

PRp = PF (1) If the tangent of the parabola A at an arbitrary point P intersects a straight line (= optical axis) B ′ passing through the focal point F and the origin O of the parabola A, then the line segment RpF And the line segment PQ have the relationship of the second equation.

RpF = PQ (2) Further, let M be a foot vertically lowered from an arbitrary point P to the optical axis B ′, and N be a point where the normal line at the point P intersects the optical axis B ′. And the line segment MN have the relationship of the third equation.

MN = 2f (= constant) (3) In the ellipse, as shown in FIG. 2, a straight line passing through _one focus F ₁ of the ellipse C having the major axis 2a and the minor axis 2b and an arbitrary point P of the ellipse C. If a point Re equal to the distance from the point P to the other focus F ₂ is taken above, the length of the line segment ReF ₁ is always constant,
There is a relation of the fourth equation between the line segment PRe and the line segment PF ₂ .

PRe = PF ₂ (4) If the point at which the tangent of the ellipse C at the point P intersects the axis X passing through the focal points F ₁ and F ₂ is Q, then between the line segment PQ and the line segment ReF ₂ There is a relationship of the fifth formula.

PQ⊥ReF ₂ (5) Furthermore, let M be the foot that descends perpendicularly from the arbitrary point P to the axis X on the focal points F ₁ and F ₂ , and the normal line at the point P is the axial line on the focal points F ₁ and F _2. If the point intersecting with X is N, then the line segment MN has the relationship of the sixth expression.

MN = x (l−e ² ) ... (6) (However, eccentricity x = distance between the origin O and the foot M. In the hyperbola, one focus of the hyperbola D as shown in FIG.
If a point Rh equal to the distance from the point P to the other focus f ₂ is taken on a straight line connecting f ₁ and an arbitrary point P of the hyperbola D, a line segment Rh
There is a relation of expression 7 between P and the line segment Pf ₂ .

RhP = Pf ₂ (7) If the point at which the connection of the hyperbola D at the point P intersects the axis X passing through the _two focal points f ₁ and f ₂ is Q, then the line segment PQ and the line segment PQ
The relation of Rhf ₂ is expressed by the equation (8).

PQ = Rhf ₂ (8) Further, let M be the foot that descends perpendicularly to the axis X on the focal points f ₁ and f ₂ from an arbitrary point P, and the normal at the point P is the axial line on the focal points f ₁ and f _2. If the point intersecting with X is N, then the line segment MN has the relationship of the expression (9).

MN = x (e ² −1) (9) (However, eccentricity e = c / a x = spacing between the origin O and the foot M. These three kinds of quadratic curves all have two specific quadratic curves. The property that the lengths of the line segments are equal, the other two specific line segments are in a right-angled relationship, and the interval between the intersection of the normal line and the axis line and the perpendicular to the axis line at any point has a constant relationship. A specific example embodying the principle having such common properties will be described below.

FIG. 4 is a schematic diagram showing an example of the present invention (parabolic drawing instrument). Reference numeral 10 denotes an L-shaped member, which is composed of a first member 11 and a second member 12 which are orthogonal to each other, and a spindle 13 is provided at each intersection position. The support shaft 13 indicates an arbitrary point P on the quadratic curve. The first member 11 is formed to extend in a direction of a point N where a normal line at an arbitrary point P intersects the optical axis, and a slide groove 11a is formed in which a slider S ₁ slides on a line connecting the arbitrary points P and N. And the second member 12 is formed tangentially at any point P and thus is formed in the form of an L-shape orthogonal to the line segment PN and on the line tangent T'T 'the slider S ₂ A slide groove 12a for sliding is formed.

The slider S ₂ that slides in the slide groove 12a of the second member 12 has the first link piece 14 that regulates the distance from the focal point F.
Is rotatably supported by a pivot line, and a quasi line D'from an arbitrary point P.
The second link piece 1 that regulates the distance between the foot D and D '
The other end of 5 is rotatably supported. First link piece 1
One end of 4 is fixed to a fixing pin 16 so as to be rotatable on a position serving as a focal point F. The other end of the second link piece 15 is rotatably supported by the slider S _2, and the other end of the second link piece 15 is a quasi line D′ D ′.
It is rotatably supported by a shaft 18 (Rp) provided at a point D of the link body 20 provided above.

The link body 20 has a T-shape for the convenience of drawing a parabola, and the point orthogonal to each other is D. The link body 20 slides in the groove 21a of the sliding groove body 21 provided on the direct line D'D ', and is arbitrary. The sliding groove body 22 is provided so as to extend on a line parallel to the optical axis connecting the points P and D. In the groove body 22, the support shaft 1 of the L-shaped member 10 is provided.
3 is slidably fitted.

The support shaft 13 is further provided with an arbitrary point P and optical axes O and N from this point P.
It is fitted in the first vertical groove 31 formed on the line connecting the foot M vertically lowered to. As shown in FIG. 5, the first groove 31 together with the second lateral groove 32 formed on the optical axis connecting the focal points F and N together with the groove plate, for example, the triangular groove plate 30 in order to regulate the orthogonal relationship. Is drilled in. The triangular groove plate 30 is provided with a support shaft for the slider S ₁ at a point N (line segment MF = 2f (= constant)) on the optical axis.

Next, the creation of a parabola using the above creation tool will be described.

First, as shown in FIG. 4, the fixing pin 16 is rotatably fixed to the drawing surface at a position to be the focus of the parabola, and
The sliding groove body 21 is arranged and fixed perpendicularly to a straight line passing through the focal point F of the parabola and the optical axis OP and parallel to the first groove 31 of the sliding groove plate 30.

In this state, a writing instrument such as a pencil, a pen or a quadratic curve generating means such as a cutter is provided at an arbitrary point P, that is, the lower end side of the support shaft 13.

Then, by sliding the link body 20 along the sliding groove body 21 in the downward direction in FIG. 4, for example, the L-shaped member 10 is formed.
Is biased counterclockwise about the fixing pin 16 (focus) by the first link piece 14 and the second link piece 15.
Along with this, the triangular groove plate 30 is displaced leftward in the drawing by the movement of the support shaft 13 (P) fitted in the first vertical groove 31 and the slider S ₁ .

In this case, the fixing pin 16 (F), the slider S ₂ and the support shaft 1
The triangle connecting 3 and the triangle connecting the support shaft 13 (P), the support shaft 18 (Rp) and the slider S ₂ maintain a congruent relationship with each other, so that the support shaft 13 and the fixing pin 16 and the support shaft 13 And the distance between the support shaft 18 and the support shaft 18 is kept constant, and a line connecting the arbitrary point P and the slider S ₁ and a line T'T 'perpendicular to PN are formed by the right angle structure of the L-shaped member 10. The connecting line maintains an orthogonal relationship, in other words, the line connecting the arbitrary point P and the slider S ₁ and the line connecting the arbitrary point P and the slider S ₂ maintain the orthogonal relationship (first,
(Formula 2) and the interval between the line segments MN are set to a constant 2f (Formula 3) as shown in FIG.
A part of the quadratic curve (parabola) A of the focal length f will be created.

Next, the creation of an ellipse using the above-described creation tool will be described. The difference from the creation of a parabola is that the parabola has an infinite second focus, so the line connecting arbitrary points P and Rp is the optical axis ON.
, That is, the ellipse has a second focal point, so that the line connecting arbitrary points P and Re is the second focal point F '. It is to cross the position that should become. Further, as described above, in the parabola, MN = 2f (= constant), whereas in the ellipse, as described above, MN = x ( ¹⁻² ), so that each of the points M, N, and O changes. Is to additionally form a pantograph 40 having nodes.

That is, as shown in FIG. 6, the fixing pin 16 is rotatably fixed at a position that should be the first focal point F of the ellipse, and the end of the link body 20 opposite to the end on which the support shaft 18 is provided (one side) The end of the pantograph 40 is rotatably supported at a position on the optical axis to be the second focal point F ', and the fixed side of the pantograph 40 is to be the origin O (a groove on the extension line of the second groove 32). It is fixed between the plate 30 and the pin position F'of the link body 20). in this case,
The movement side linkage of the pantograph 40 is in the vertical groove 31 (M).
Since it is located at, it is necessary to attach it to a support shaft provided below the triangular groove plate 30 via a metal fitting.

In this state, a pencil is attached to an arbitrary point P, that is, the lower end of the support shaft 13,
A writing instrument such as a pen or a cutter such as a cutter should be installed.

Then, the L-shaped member 10 is moved to the first position by rotating the link body 20 clockwise around the position where the second focal point F'is to be set.
The slider 13 is rotated clockwise by the restriction of the link piece 14 and the second link piece 15, and the triangular groove plate 30 is fitted with the support shaft 13 (P) fitted in the vertical groove 31 and the lateral groove 32.
It is biased to the right in the figure by the movement on S ₁ .

In this case, the fixing pin 16 (F), the slider S ₂ and the support shaft 1
Triangle connecting 3 (P), support shaft 13 (P), support shaft 18 (R
e) and the triangle that connects the slider S ₂ maintain a congruent relationship with each other, so that the distance between the support shaft 13 and the fixing pin 16 and the support shaft 13 and the support shaft 18 is kept constant, and the L-shaped Since the line connecting the arbitrary point P and the slider S ₁ and the line connecting the arbitrary point P and the slider S ₂ maintain the orthogonal relationship by the right angle structure of the member 10 (the fourth and fifth equations), and Since the interval between the points M and N is set in the pantograph 40 so that the interval is scaled similarly (equation 6), a part of the ellipse B is created.

In the example of FIG. 6, as the point P is separated from the vertical axis passing through the origin O (x becomes smaller), the point M approaches the position of the fixing pin 16 at the focal position, but the point M is the pin 16. It is possible to draw until it touches.

Next, the creation of a hyperbola using the above-mentioned creation tool will be described. The difference from the creation of a parabola is that the parabola has an infinite second focus, so the line connecting arbitrary points P and Rp is the optical axis ON.
The hyperbola must be installed parallel to the second
Since the eccentricity e is 1 or more in the second focus, the L-shaped member 10 and the link pieces 14 and 15 are used by being inverted, and the support shaft 18 is used as the slider restricting groove of the link body 20. 22 and the fixing pin F'on the one end side 20a to rotatably support the one end of the second link piece 15.
The line connecting the arbitrary points P and Rh is to intersect with the second focal point F '. As described above, in the parabola, MN = 2f (= constant), whereas in the hyperbola, as described above, NM = x (e ² −1). That is, the pantograph 50 having each node as a node is additionally formed.

That is, as shown in FIG. 7, the fixing pin 16 is rotatably fixed to a position where the second focal point F of the hyperbola should be formed, and the one end 20a of the link body 20 is moved to the first position on the optical axis. It is rotatably fixed to a position to be the focal point F ', and the node on the fixed side of the pantograph 50 is to be the origin O (the groove plate 30 and the link body on the extension line (optical axis) of the second groove 32). It is fixed at a position between one end side 20a (F ') of 20).

In this state, a pencil is attached to an arbitrary point P, that is, the lower end of the support shaft 13,
A writing instrument such as a pen or a cutter such as a cutter should be installed.

Then, by rotating the link body 20 counterclockwise about the first focal point F ′, the L-shaped member 10 is rotated about the support shaft 13 by the restriction of the first link piece 14 and the second link piece 15. While rotating in the clockwise direction, the triangular groove plate 30 is biased to the right in the drawing by the movement on the support shaft 13 fitted in the vertical groove 31 and the slider S ₁ fitted in the lateral groove 32. In this case, the triangle connecting the fixing pin 16, the slider S ₂ and the support shaft 13 and the triangle connecting the support shaft 13, the support shaft 18 and the slider S ₂ maintain a congruent relationship with each other, and thus the support shaft 13 and The distance between the fixing pin 16 and the support shaft 13 and the support shaft 18 is kept constant, and moreover, depending on the structure of the L-shaped member 10, an arbitrary point P
And the slider S ₁ and an arbitrary point P, the line connecting the slider S ₂ maintains an orthogonal relationship (equations 7 and 8), and the distance between the points M and N is the pantograph 50. Are set so as to be scaled in a similar manner (Equation 9), so that a part of the hyperbola C is created. Therefore, a corresponding hyperbola C'is also created by inverting and utilizing the first and second focal points.

In FIG. 7, the point M approaches the position of the fixing pin 16 at the focal position as the point P moves away from the vertical axis passing through the origin (x becomes larger), but the point M becomes closer to the fixing pin 16. It is possible to draw until it touches.

As described above, according to the present invention, the L-shaped member composed of the first and second members having a right angle is formed, and the slider restricting groove is formed in the first and second members. The first slipper is fitted in the slipper restricting groove of the member, the second slide is fitted in the slipper restricting groove of the second member, and two orthogonal grooves are provided in the groove plate. A pin planted at a right-angled intersection point of the L-shaped member is fitted in the groove, and the other end of the first and second link pieces and the second slider are rotatably rotatably supported. In the next curve creation tool, a spindle is planted on the extension line of the second groove of the groove plate,
And the second groove is slidably provided to one end of the first link piece and a fixing pin that is rotatably supported. The sliding groove body is provided at a position parallel to the groove, and the link body that fits in the sliding groove body and slides while keeping the parallel state with the second groove is provided. Then, the pin of the L-shaped member is fitted in the pin regulating groove, the shaft is provided on the link body at the intersection of the link body and the sliding groove body, and the shaft and one end of the second link body are rotatable. An L-shaped member having a pivotally supported structure and a right-angled first member and a second member is formed, and a slider restricting groove is formed in each of the first and second members to restrict a slider of the first member. The first slide is fitted in the groove, the second slide is fitted in the slide regulating groove of the second member, and the second slide is fitted in the groove plate.
Two orthogonal grooves are provided, and a pin planted at a right-angled intersection position of the L-shaped member is fitted into the first groove, and the other end of the first and second link pieces and the second slider In a quadratic curve creating tool that rotatably pivots, a first slide and a rotatably provided pin are fitted in the second groove of the groove plate, and the second groove is The link piece is slidable with respect to the fixing pin rotatably supported at its one end, and a linear link body is formed with the fixing pin at one end, and the link body has a slider regulating groove. Is provided, the slider is fitted in the slider restriction groove, the slider of the link body and the pin of the L-shaped member are rotatably supported, and the other end of the link body is the second end.
Is rotatably supported on one end of the link piece and is linked to the groove plate at the intersection position of the first and second grooves, the rotation center position of the first slider and the extension line of the second groove. A configuration in which a pantograph that deforms similarly with a position between the body and a pin as a node is provided, and an L-shaped member including a right-angled first and second member is formed, and a slip is formed on the first and second members. A child restricting groove is formed, a first slider is fitted in the slider restricting groove of the first member, and a second slider is fitted in the slider restricting groove of the second member. Providing orthogonal grooves, the first groove is fitted with a pin planted at a right-angled intersection point of the L-shaped member, and the other end of the second link piece having the same length and the second slider are attached. In a quadratic curve generating tool rotatably supported, a first slider and a rotatably provided pin are fitted in the second groove of the groove plate, and the second groove is formed of the first phosphorus. A straight link body is provided that is slidable with respect to a fixing pin that is rotatably supported at one end, and a fixing pin is provided at one end, and a slider restricting groove is provided on the link body. The slider is fitted in the slider restriction groove, the slider of the link body and the pin of the L-shaped member are rotatably supported, and the slider is planted between the slider restriction groove of the link body and the one end side fixing pin. One end of the second link piece is rotatably supported on the provided support shaft, and the rotation center position of the first slider, the intersection point position of the first and second grooves, and the extension of the second groove Since a pantograph that deforms similarly with the position between the fixing pin of the groove body and the fixing pin of the link body as a node on the line is provided, it is possible to easily perform various types of quadratic curves without using various auxiliary lines as in the past. Since it is possible to create a quadratic curve while understanding the properties of, it is ideal for teaching and requires less configuration than before. Since it is very small, it is easy to manufacture, and it is possible to form, polish, and grind various lenses and mirror surfaces of large-scale optical systems, and it can be supplied at low cost when applied to teaching materials. .

[Brief description of drawings]

FIG. 1 is a diagram for explaining a parabola of a quadratic curve, FIG. 2 is a diagram for explaining an ellipse of a quadratic curve, FIG. 3 is a diagram for explaining a hyperbola of a quadratic curve, and FIG. The top view which creates a parabola which shows an example of an invention creation tool, FIG. 5 is a plan view of a triangular groove board, FIG. 6 is explanatory drawing which creates an ellipse using the creation tool of this invention, FIG. 7 is this invention. It is an explanatory plan view which creates a hyperbola using a creation tool. 10 ... L-shaped member, 11 ... First member, 12 ... Second member, 13 ... Spindle, 14 ... First link piece, 15 ...
Second link piece, 20 ... Link body, 30 ... Triangular groove plate, 31 ... First vertical groove, 32 ... Second horizontal groove.

Claims (3)

[Claims] 1. An L-shaped member composed of a right-angled first member and a second member is formed, and a slipper restricting groove is formed in each of the first and second members. Is the first slide,
The second slider is fitted into the slider restriction groove of the second member,
The groove plate is provided with two orthogonal grooves, and the first groove is fitted with a pin planted at a right-angled intersection position of the L-shaped member, and the other end of the first and second link pieces and the second link piece. In a quadratic curve generating tool that pivotally supports a slider of the above, a support shaft is planted on an extension line of the second groove of the groove plate, and the support shaft and the first support
And the second groove is provided slidably with respect to the one direction end of the first link piece and a fixing pin rotatably rotatably supported. The sliding groove body is provided at a position parallel to the first groove, and the link body that fits in the sliding groove body and slides while maintaining the parallel state with the second groove is provided. Formed, fit the pin of the L-shaped member in the pin restriction groove, provide a shaft on the link body at the intersection of the link body and the sliding groove body, and freely rotate one end of the shaft and the second link body. A quadratic curve creation tool characterized by being pivotally supported on. 2. An L-shaped member composed of a right-angled first member and a second member is formed, and a slipper restricting groove is formed in each of the first and second members, and a slipper restricting groove of the first member is formed. Is fitted with a first slider, the second member is fitted with a second slider in a slider restriction groove of the second member, two orthogonal grooves are provided on the groove plate, and the first groove is provided with an L-shaped member. A quadratic curve forming tool in which a pin implanted at a right-angled intersection is fitted, and the other ends of the first and second length link pieces and the second slider are rotatably supported. A pin that is rotatably provided with the first slider is fitted in the second groove of the plate, and the second groove is fixed such that the first link piece is rotatably supported at one end thereof. A linear link body that is slidable with respect to the work pin and has a fixing pin at one end is formed.The link body is provided with a slipper restricting groove, and the slider is fitted in the slipper restricting groove. Of the slider and the pin of the L-shaped member are rotatably supported, and the other end of the link body rotatably supports one end of the second link piece. Is provided with a pantograph that deforms similarly with the intersection point position, the rotation center position of the first slider and the position between the groove plate and the pin of the link body on the extension line of the second groove as nodes. Next curve generator. 3. An L-shaped member composed of a right-angled first and second member is formed, and a slipper restricting groove is formed in the first and second members, and the slipper restricting groove of the first member is formed. The first slide,
The second slider is fitted into the slider restriction groove of the second member,
The groove plate is provided with two orthogonal grooves, and the first groove is fitted with a pin planted at a right-angled intersection position of the L-shaped member, and the other end of the first and second link pieces and the second link piece. In a quadratic curve generating tool rotatably rotatably supporting the first slider and the rotatably provided pin, the second groove is formed in the second groove of the groove plate. Is slidable with respect to a fixing pin that is rotatably supported at one end of the first link piece, and forms a linear link body having a fixing pin at one end. A restriction groove is provided, and the slider is fitted in the slider restriction groove, and the slider of the link body and the pin of the L-shaped member are rotatably pivotally supported to fix the slider restriction groove of the link body and the one end side. One end of the second link piece is rotatably supported by a support shaft that is planted between the pin and the rotation center position of the first slider, and the intersection position of the first and second grooves. Beauty quadratic curve creating device, characterized in that a pantograph to similarity transformation as a node position between the fixing pins of the second groove member in an extension of the groove and the link member.