Parametric equation

Under a parametric representation (also parameterization or configuration) is understood in mathematics, a representation in which the points of a curve or surface are as a function of one or more variables, the parameters passed. For the description of a curve in the plane or in space, a parameter is needed to describe a face, a set of two parameters.

An example is the description of the unit circle to the origin of a Cartesian coordinate system in the plane. One possible parameter is the angle at the origin (see adjacent picture), thus obtaining the following parametric representation of the position vector as a function of:

The description of the path coordinates of a moving object as a function of time is an example of a parametric representation in physics.

If a parametric representation of a curve or surface is known, the corresponding point on the curve or surface can be specified for each parameter (set) directly. However, it is usually difficult to determine whether a given point is on the curve or surface.

Curves or surfaces can be parameterized in different ways. At curves it is often convenient to choose the arc length as measured from a fixed point along the curve as a parameter. The parameters of surfaces and higher dimensional structures are often selected so that the parameter lines are orthogonal. Even with relatively simple structures, it is not always possible to find for each parameterization a parametric representation of the coordinates with the aid of elementary functions, such as when in an ellipse, the arc length is selected as a parameter.

Properties of the parametric equations

In addition to the parametric representation, there are also other ways to describe curves or surfaces. In the plane, for example, describes the graph of a function of a curve in three-dimensional space can be described by the function of a surface. These are special parametric equations, if one conceives the function variables as parameters. However, they are not suitable for the representation of figures such as circles or spheres, as they each point of the axis, or - can only assign a point level. With the function

Only one semicircle are presented. To obtain a full circle, a semicircle must be further added.

Another possible is the implicit description by an equation of the coordinates, for example. The unit circle can be in this form by the circle equation

Describe. This form is a good way to check whether a given point lies on a curve or level, since only necessary to check whether the coordinates satisfy the equation. With such implicit equation only objects can be described, whose dimension is less by 1 than those of the space in which they are described. An equation that extends in three-dimensional space to describe an area, but not to describe curves.

In a parametric representation, it is easy to calculate individual points that belong to the parametric curve or surface. It is therefore well suited to draw these objects, for example in CAD ​​systems. In addition, the calculated coordinates can be easily transformed into other coordinates, so that objects relatively easily moved, can be rotated or scaled.

In physics, the parameter representation is used to describe the path of moving objects, where usually the time is chosen as a parameter. The derivative of the position vector with respect to time is then given the time-dependent speed, the second derivative of the acceleration. Conversely, if an initial position and initial velocity at the time and a (possibly space-and time -dependent ) where acceleration field, we obtain the parametric representation of the path curve by integration. At a constant acceleration as in the oblique throw without air resistance, for example, gives the following trajectory:

Parametric equations are also used in differential geometry. With the help of derivatives of the position vectors according to the parameters can be lengths, tangent vectors and tangent planes, determine curvatures, angles or surface content. For the calculation of length, angles and surface areas in the surfaces, it is not necessary to know an explicit representation of the parameter area in the room. It is sufficient if the metric ( first fundamental form ) of the surface, which describes the lengths of the parameter along lines, and the angle between the line parameter is known. This may be advantageous on curved surfaces.

Parametric equations of lines and planes

Under the parametric representation (or parameters form) a linear equation refers to the form

And a plane equation in the form

Where and are the real parameters. The vector is the position vector of a point on the straight line or plane. This point is called the receptor point or base, its position vector is then called support vector. The vector in the linear equation is called the direction vector of the line, the vectors in the plane equation and also the direction vectors or tension vectors. These vectors may no zero vectors, the tension vectors of a plane also not be collinear. If the linear equation is a unit vector corresponding to the parameters of the distance of a point of line.

The direction vectors of a plane equation span an affine coordinate system on ( in the picture by the blue coordinate grid within the plane indicated ) representing where and are the affine coordinates. The position vector of a point of the plane is obtained by adding the position vector of the point, the times of the vector, and then the times of the vector.

Regular parametric equations

A differentiable parametric representation of a curve is called regular if its derivative vanishes at any point; it need not necessarily be injective. General ie a differentiable parametric representation regular if it is an immersion, that is, when its derivative is everywhere injective ( that is, its rank is greater than or equal to the dimension of the preimage ).

Generalization to higher dimension

The generalization is obvious: There is a " map " of a -dimensional differentiable manifold. The map is given by a -dimensional differentiable parameterization: For points in thus applies: with differentiable functions.

Then for an arbitrary function of the points of the manifold on for the derivative in the direction of the tangent vector of a curve on the map the curve parameter λ:

This result is due to the chain rule, regardless of the chosen parameterization.