Pedal curve

The base point of transformation is in mathematics an operation that forms from a curve in the plane a new curve, their Fußpunktkurve.

Mathematical representation

For the construction of the in-plane Fußpunktkurve a point (called the pin) is selected. A given curve is then mapped as follows: a point is the foot of the perpendicular from the tangent of in allocated.

The construction of the image point can be elementarily describe: is the intersection of the tangent to the circle of Thales in about. The tangent to the Fußpunktkurve in is the tangent to the circle of Thales in. This also provides the important insight that not the entire curve must be known in order to construct the pixel, but only the point itself and the direction of the tangent.

The construction of the image point can be described analytically: We attach to a Cartesian coordinate system through the pole and think we are given the point by coordinates. The tangent direction is defined by. Wanted are now the coordinates of the base point. We will continue to determine the direction of the tangent Fußpunktkurve in.

Since the point is situated on the tangent to pass and on the normal through the coordinates satisfy the equations

As a result and. Furthermore, it can be determined with the differential calculus:

Properties

Examples

The following are examples of Fußpunktkurven be considered. Here, the term " curve" to be understood in a wider sense, eg let a point be understood as a curve.

• Straight

• Circles

• Points

• Parabolas, conics

Conservation of line elements

In mathematics, a triple is called a line element. The analytical formulas of the base point transformation show that the line elements are mapped one-one with each other.

Yourself by touching two curves ( ie they have a point next to the tangent together ), then the Fußpunktkurven touch in each pixel.

Importance

Since the base point transformation line elements mapping is one-one each other, they can be exploited as a " transfer principle " in the sense of Klein's Erlanger Program: For certain propositions about points, lines and conics can be directly theorems about points, lines and circles prove and vice versa. Some examples of sentences that can be transmitted by applying the base point transformation: