 Mathematics Education

## Envelopes of Lines and Circles

### J. W. Wilson

#### Click here for a GSP animation of the outline of a parabola by the envelope of its tangent lines.

The construction of the parabola begins with a fixed point for a focus and a line for the directrix. The set of points equidistant from the focus and the directrix is the parabola. From a random (variable) point on the directrix draw a perpendicular. A point on this perpendicular would be equidistant from the directrix and the focus if an isosceles triangle was formed. The altitude of this isosceles triangle would line on a tangent line to the parabola at the constructed point.  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.

2. An Astroid.

Note:  The mathematical curve with four curved sides is an ASTROID.    The object flying through spacemay be an ASTEROID. The astroid is outlined by a trace of segments of fixed length so that one end travels along the y-axis and the other end moves along the x-axis, either to the left or to the right.

This astroid is rectangular in that its major axes are perpendicular. A modified figure is obtained by taking oblique axes rather than perpendicular axes.

To explore the envelope of lines for the Astroid with a GSP sketch, click here. Consider the locus of the midpoints of the two line segments that generate the astroid.  Click herefor a GSP sketch. The locus is a circle with center at the intersection of the axes and tangent to the curved "sides" of the astroid  (or inscribed in the astroid)  This is the case when the axes are orthogonal. When the axes are oblique, the locus of the midpoints is an ellipse with center at the intersection of the axes and inscribed in the oblique astroid. The envelope of circles with center at one of these midpoints and passing through the fixed point at the origin (or having the segment as a diameter) can be easily drawn. Click here for a GSP sketch.

The envelope of circles creates a disc circumscribing the astroid. The oblique case shows something more with the envelope of circles.    We see the the astroid is circumscribed by an ellipse, as might be expected, but there is an additional astrod created that is circumscribed by the ellipse of midpoints. Note that our original astroid was from an envelope of lines whereas this new astroid is from the envelope of circles. 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 astroid, the locus will be an ellipse inscribed in the astroid.

The Astroid is also the locus of a point on a small circle  of radius  r  rolling around the inside of a larger circle of radius   R where    R  =  4r.     That is, it is a hypocycloid of four cusps.    Click HERE for an animation. Some Cartesian equations for the Astroid are the following: The graph of either is shown at the right.

#### The Ellipse Consider a circle with a fixed point inside, but not the center, and a random point on the circle. The ellipse is the set of points equidistant from the fixed point and from the circle. Note this is description is similar to the description of a parabola except that the directrix is now a circle rather than a line. The distance from the circle is along a line through the center of the circle (the distance from a curve at a point is along the perpendicular to the tangent to the curve at the point). 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.

#### The Hyperbola The construction for the ellipse, with the circle as a directrix, can be repeated with the fixed point outside the circle. The result is an envelope of tangent lines of the hyperbola with foci at the fixed point and the center of the circle.

#### The lemniscate An interesting extension for the hyperbola is to consider an envelope of circles generated by taking a variable point on the hyperbola as the center of a circle that passes through the midpoint of the segment connecting the foci of the hyperbola 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 the circles. 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.

#### Envelopes of Circles generated by an Ellipse

Take a variable point on an ellipse and a fixed point not on the ellipse. Using the variable point on the ellipse as the center of a circle that passes through the fixed point, an envelope of circles is traced as the variable point moves around the ellipse. When the fixed point is on the orthogonal to the center of the segment connecting the two loci of the ellipse, the following figure results.

Click HERE for a GSP sketch to experiment with various locations of the fixed point. The above figure is symmetric. When the fixed point is moved to a point above the line of foci of the ellipse and to the left of the perpendicular to the center of the line of foci, then the envelope generates a non-symmetric figure. Another symmetric figure results by placing the fixed point on the line of the foci. 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 a limaçon. If the fixed point is on the circle, the resulting envelope is a cardioid:.

The Nephroid

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.

The Nephroid is also an Epocycloid of two cusps.  The curve is the locus of a point on a circle of radius   r   rolling around the ouside of a circle of radius   R where  R = 2r.