The deltoid is the locus of a point on the circumference of a circle that
rolls inside of a fixed circle whose diameter is three times as large.
Construct an animation showing how the deltoid is formed.
The astroid is the locus of a point on the circumference of a circle that
rolls inside of a fixed circle whose diameter is four times as large.
Construct an animation showing how the astroid is formed.
The nephroid is the locus of a point on the circumference of a circle that
rolls outside of a fixed circle whose diameter is twice as large. Construct an
animation showing how the nephroid is formed.
The cardioid is the locus of a point on the circumference of a circle that
rolls outside of a fixed circle with the same diameter. Construct an animation
showing how the cardioid is formed.
In the construction of the astroid as the envelope of its tangents, find the
point of tangency.
In the construction of the cardioid as the envelope of circles, find the point
of tangency.
In the construction of the nephroid as the envelope of circles, find the point
of tangency.
In the construction of the deltoid as the envelope of straight lines, find the
point of tangency.
When the cardioid is formed as an envelope of line segments joining
with
on the unit circle, where is the point of tangency?
When the neohroid is formed as an envelope of line segments joining
with
on the unit circle, where is the point of tangency?
The feet of perpendiculars to the sides of a triangle from a point are
collinear if and only if the point is on the circumcircle of the triangle. In
this case, the line through the feet of perpendiculars to the sides of the
triangle is called the Simson line of the point with respect to the
triangle.
What is the envelope formed by the Simson lines of a fixed
triangle?