Base point transformation

In mathematics, the base point transformation is an operation that creates a new curve, its base point curve, from a curve in the plane.

Mathematical representation

Base point transformation

For the construction of the base point curve, a point (the so-called pole ) is selected in the plane . A given curve is then mapped as follows: A point is assigned the base point of the perpendicular from on to the tangent from in . ${\displaystyle O}$ ${\displaystyle c}$ ${\displaystyle P\in c}$ ${\displaystyle F}$ ${\displaystyle O}$ ${\displaystyle c}$ ${\displaystyle P}$

The construction of the image point can be described elementarily: is the point of intersection of the tangent to in with the Thales circle over . The tangent to the base point curve in is the tangent to the Thales circle in . This also results in the important insight that not the entire curve has to be known in order to construct the image point, but only the point itself and the direction of the tangent. ${\displaystyle F}$ ${\displaystyle c}$ ${\displaystyle P}$ ${\displaystyle {\overline {OP}}}$ ${\displaystyle F}$ ${\displaystyle F}$

The construction of the image point can be described analytically: We place a Cartesian coordinate system through the pole and think of the point as given by coordinates . The tangent direction is determined by. We are now looking for the coordinates of the base point . We will also determine the tangent direction of the base point curve in . ${\displaystyle O}$ ${\displaystyle P}$ ${\displaystyle (x,y)}$ ${\displaystyle y'}$ ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle F}$ ${\displaystyle y_{1}'}$ ${\displaystyle F}$

Since the point lies on the tangent to through as well as on the normal through , its coordinates satisfy the equations ${\displaystyle F}$ ${\displaystyle c}$ ${\displaystyle P}$ ${\displaystyle O}$ ${\displaystyle (x_{1},y_{1})}$

${\displaystyle (x_{1}-x)y'-(y_{1}-y)=0}$

${\displaystyle x_{1}+y_{1}y'=0}$

This results in

${\displaystyle x_{1}=-{\frac {y'(y-xy')}{1+y'^{2}}}}$

and

${\displaystyle y_{1}=-{\frac {(y-xy')}{1+y'^{2}}}}$.

Furthermore, the differential calculus can be used to determine: ${\displaystyle y_{1}'}$

${\displaystyle y_{1}'={\frac {\mathrm {d} y_{1}}{\mathrm {d} x_{1}}}={\frac {xy'^{2}-x-2yy'}{yy'^{2}-x+2xy'}}}$

properties

Examples

In the following, the term "curve" is to be understood in a broader sense, e.g. B. a point should also be understood as a curve.

• Straight lines

The base point curve of a straight line is a point: For every point on the straight line the tangent is this straight line itself. There is exactly one base point of the perpendicular of the straight line through the pole . ${\displaystyle O}$

• Circles

The base point curve of a circle whose center is the pole is the circle itself. If the pole is different from the center of the circle, the base point curves are more complicated. ${\displaystyle O}$

• Points

The base curve of a point is the circle with as diameter. Tangents to a point are all possible straight lines through that point. That this definition makes sense can be explained by considering points as "degenerate circles". ${\displaystyle P}$ ${\displaystyle {\overline {OP}}}$

• Parabolas, conic sections

The base curve of a parabola with the pole as the focal point is the tangent to the parabola through its vertex. In general, conic sections with the pole as the focal point are mapped onto circles whose diameter is the main axis of the conic section. ${\displaystyle O}$ ${\displaystyle O}$

Preservation of line elements

In mathematics, a triple is called a line element . The analytical formulas of the base point transformation show that line elements are uniquely mapped onto one another. ${\displaystyle (x,y,y')}$

If two curves touch each other (i.e. they have a point and the tangent in common), the base point curves touch each other in the image point.

meaning

Since the base point transformation maps line elements uniquely to one another, it can be used as a "transfer principle" in the sense of Klein's Erlanger program : Certain sentences about points, straight lines and conic sections can be used to directly prove sentences about points, straight lines and circles and vice versa. Some examples of sentences that can be transferred by applying the foot point transformation:

Base point transformation as a transfer principle
Theorems about points, lines and conic sections	Theorems about points, lines and circles
Two points define a straight line	Two intersecting circles that have one point in common have another point in common.
Two straight lines intersect at one point	Three points define one and only one circle.
A conic section is clearly defined by a focal point and three tangents.	Three points define one and only one circle.
There are eight common focus conics that touch three conics with the same common focus ${\displaystyle O}$	There are eight circles that touch three given circles.

literature

• Sophus Lie and Georg Scheffers: Geometry of Touch Transformations. Chelsea Publishing Company, ISBN 0-8284-0291-4