Thus the envelope of tangent lines for the parabola can be generated
by tracing the image of the perpendicular to the segment between the focus
and a variable point on the line.
This asteroid is rectangular in that its major axes are perpendicular.
A modified figured is obtained by taking oblique axes rather than perpendicular
Explore the envelope of lines for the Asteroid with a GSP sketch. Click
Consider the locus of the midpoints of the two line segments generating that move to generate the asteroid. The envelope of circles with center at one of these midpoints and passing through the fixed point at the origin can be easily drawn. Click here for a GSP sketch.
The picture suggests that the locus of the midpoint is an inscribed circle
in the asteroid and the envelope of circles defines a circumscribed circle
of the asteroid. However, when oblique axes are used, the locus of the midpoints
is an inscribed ellipse.
If points not at the center of the segments are selected and the locus
of those points traced as the envelope of lines forms the asteroid, the
locus will be an ellipse inscribed in the asteroid.
Click here for a movie that illustrates
the remarks above.
As in the case of the parabola, an isosceles triangle is formed and the
line along the altitude is a tangent to the ellipse. Tracing the tangent
lines as the variable point moves around the circle gives an envelope of
lines to outline the ellipse.
Now, as the variable point (center of the circle) moves on the hyperbola,
an envelope of circles is generated. Click here
to see a GSP animation of this.
The envelope identifies a lemniscate. The lemniscate is the inversion
of the hyperbola in circle. Thus the construction of an envelope of circles
accomplishes an inversion of the curve that is the path of the centers of
Moving the fixed point of the envelope of circles to some point other than the midpoint of the segment connecting the loci of the hyperbola produces a modified "lemniscate."
To experiment with a GSP sketch where other "centers" can be
explored, click here or click here
for a movie illustrating the same idea. For example, locating the fixed
point of the envelope at either foci generates an envelope that defines
a circle. A kidney shape results when the fixed point is on the hyperbola
but not on the line segment joining the foci.
Another symmetric figure results by placing the fixed point on the line
of the loci:
Special cases of this figure result when the ellipse is replaced by a
circle. If the fixed point is outside the circle, the resulting figure is
If the fixed point is on the circle, the resulting envelope is a cardioid:.
The nephroid is formed by an envelope of circles with centers on a given circle and each tangent to a diameter of the circle.
Click here to obtain a GSP script for
generating the nephroid. Interesting variations occur when the line is a
chord other than a diameter:
or a tangent line:
or a line that does not intersect or touch the circle:
Click here for an animations of these
transformations of the envelope of circles.
Next, the nephroid is outlined in two ways -- as an envelope of circles and as an envelope of lines.
A GSP sketch to allow experimenting with these envelopes, click here.