Mathematics with Cinderella: Various Curves: Strophoid
We will draw a strophoid using the construction method described in "Encyclopedia of Curves" (Masami Isoda et al., Kyoritsu Shuppan, 2009).
Definition of a Strophoid
Given a fixed point O and a moving point P on a line s, and a point A not on line s. As point P moves along line s, the locus of points M1 and M2 on line AP that satisfy PM1 = PM2 = OP is called a strophoid (leaf-shaped curve).
Since it is a simple construction, let's try it. You can set line s as either the x-axis or the y-axis. The following figure uses the y-axis. The names of the points have been changed after construction to match the definition.
Newton's Construction Device
In Newton's construction device, a strophoid is drawn along with a cissoid. The following figure shows a strophoid drawn using the same device as the cissoid construction device.
I have posted a simulation of the construction device on the next page.
Quetelet's Construction Device
Two spheres of radius R are inscribed in a cylinder (with centers O and O'). When the cylinder is cut by a plane tangent to the two spheres, let F and F' be the points of tangency between the cutting plane and the spheres, and let P and Q be the vertices of the ellipse obtained as the cross-section. Here, consider the cross-section of the cylinder by a plane passing through the three points O, O', and P. As shown in the figure, the cross-section consists of two parallel lines a and b, a circle with center O or O' tangent to these two lines, and a line PQ tangent at the two points F and F'. When the two circles roll within the horizontal region bounded by the two lines a and b, and the line PQ rotates around point P while remaining tangent to the two circles, the two points F and F' trace a strophoid.
Although it mentions "spheres" and a "cylinder," they are actually disks as shown in the photo. Photos on the web can be found on the next page.
Therefore, you can construct it as shown in the figure on the right. Let P be a fixed point and Q be a moving point, draw two circles such that PQ is a common tangent, and draw the locus of their points of tangency.
① Draw three parallel lines.
② Draw a diagonal line passing through a point on the bottom line and a point on the top line.
Let the bottom point be the fixed point and the top point be the moving point.
③ Draw the angle bisectors of the angle with the bottom line. Two lines, one to the right and one to the left.
④ Find the intersection with the center line.
⑤ Draw a circle centered at the intersection from ④ that is tangent to the top and bottom lines.
⑥ Draw perpendicular lines from the centers of the two circles to the diagonal line and find the intersection points.
Regarding the circle in ⑤, if you move point H in the figure onto the y-axis in the background, the angle bisector will be at a 45-degree angle, so the center will fall on a grid point. You can set the radius well by using the snap function. Alternatively, you can draw a circle with a fixed radius.
Using the locus tool, if you set the moving point as H and the point that draws the locus as N or M, a strophoid will be drawn.
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