Definitions

Caustic

Caustic is a method of deriving a new curve based on a given curve and a point. A curve derived this way may also be called caustic. Given a curve C and a fixed point S (the light source), catacaustic is the envelope of light rays coming from S and reflected from the curve C. Diacaustic is the envelope of refracted rays. Light rays may also be parallel, as when the light source is at infinity.

Caustic do not always generates a curve. For example, the light rays reflected from a parabola's focus do not intersect, therefore its envelope do not form any curve.

Envelope

Envelope can be thought of as a way of deriving a new curve based on a set of curves. The envelope of a set of curves is a curve C such that C is tangent to every member of the set. (Two curves are tangent to each other iff both curves share a commen tangent at a common point.)

The concept of envelope is easily understood by looking at its graph. When a family of curves are drawn together, their envelope takes shape.

Evolute

Evolute is a method of deriving a new curve based on a given curve. It is the locus of the centers of osculating circles of the given curve.

Involute

Involute is a method to derive a new curve based on a directioned curve and a point on the curve. It is the roulette of a selected point on a line that rolls (as a tangent) upon the curve.

Step by step description:

1.Given a curve (a unit circle for example) and a point O on the curve.
2.Imagine a point P on the curve starting from O and moving through the curve.
3.At each moment in time, let there be a tangent at P.
4.Mark a point Q on this tangent such that the arc length from O to P is equal the line length P to Q. (there are two such points, take the one that lie "behind" P)
5.The locus of Q is the involute of the given directioned curve with respect to the point O on the curve.

Inversion

Inversion in geometry is a transformation. Let P be a given point. Let c be a circle centered on O and radius r. The inverse of P with respect to c is a point Q on the radial line[O,P] such that distance[O,P] * distance[O,Q] == r^2. From this definition, two properties can be seen easily: (1) Q is an inverse of P if and only if P is an inverse of Q. (2) Points inside the circle are mapped to the outside and vice versa. Points on the circle are fixed points (i.e. the inversion of any point on the inversion circle is itself). As P moves futher from O, its image Q moves closer to O. From this observation, we may then define the inverse of a point on the center of the inversion circle to be a point at infinity, and vice versa. With such a definition, we have obtained a transformation on a plane that has a point at infinity. This concept is important in geometry, and is used in stereographic projection. It shows that such a plane is topologically equivalent to a sphere. Also, inversion can be thought of as a generalization of reflection. A normal reflection is then an inversion with the inversion circle radius infinitly large.

The inversion of a curve is the inverse of all points on the curve. It can be thought of as a way to derive a new curve based on a given curve and a circle. If curve A is the inverse of curve B, then curve B is also the inverse of curve A with respect to the same circle. The center of the inversion circle is called the pole. One property easily seen from the definition is that the radius of inversion circle effects the scale of the inversion curve, but does not effect its shape. Curves that inverts into themselves are called anallagmatic curves. Circles, lines, and Cassinian ovals are anallagmatic curves. Asymptotes to a curve C invert into a curve that is tangent C's inverse.

Step by step description:

1.Given a curve C, and a circle M centered at O with radius r.
2.Draw a line passing O and any point P on the curve.
3.Mark a point Q on this line such that distance[O,P] * distance[O,Q] == r^2.
4.Repeat this for other points P on the curve.
5.The locus of Q is the inverse of curve C with respect to circle M

Pedal and negative pedal

Pedal and negative pedal are methods of deriving a new curve based on a given curve and a point.

Step by step description for positive pedal:

1.Given a curve and a fixed point O.
2.Draw a tangent at any point P on the curve.
3.Mark a point Q on this tangent so that line PQ and line OQ are penpendicular.
4.Repeat step (2) and (3) for every point P on the curve. The locus of Q is the pedal of the given curve with respect to point O.

Pedal and negative pedal are inverse concepts. Negative pedal of a curve C can be defined as a curve C' such that the pedal of C is C'.

Step by step description for negative pedal:

1.Give a curve and a fixed point O.
2.Draw a line from O to any point P on the curve.
3.Draw a line perpendicular to line OP and passing P.
4.Repeat step (2) and (3) for every point P on the curve. The envelope of lines is the negative pedal of the given curve with respect to point O.