plot::Polar – curves in 2D parameterized in polar coordinates

plot::Polar creates parameterized curves in 2D, with parametrization in polar coordinates.

Call:

plot::Polar([r, ], u = $`u_{min}`$ .. $`u_{max}`$ , <a = amin .. amax>, Options)

Parameters:

r, :	the coordinate functions: arithmetical expressions or piecewise objects depending on the curve parameter and the animation parameter a. Alternatively, MuPAD procedures that accept 1 input parameter or 2 input parameters , and return a real numerical value when the input parameters are numerical. r, are equivalent to the attributes XFunction, YFunction.
u:	the curve parameter: an identifier or an indexed identifier. u is equivalent to the attribute UName.
$`u_{min}`$ .. $`u_{max}`$ :	the plot range for the parameter : $`u_{min}`$ , $`u_{max}`$ must be numerical real values or expressions of the animation parameter . $`u_{min}`$ .. $`u_{max}`$ is equivalent to the attributes URange, UMin, UMax.

Details:

plot::Polar creates curves in one parameter, with parametrization in polar coordinates and possibly animated (see examples 1 and 2). The curves may contain poles, in which case automatic clipping is used by default, see example 4.
Polar coordinates consist of a radius and an angle. The radius of a point is its distance from the origin , while the angle is the angle between the positive “”-axis (the ordinate) and the connection between the point and the origin, measured in radians and counter-clockwise.
By default, curves are sampled at equidistant values of the parameter t. The attribute AdaptiveMesh can be used to change this behavior, such that a denser sampling rate is used in areas of higher curvature. Cf. example 3.
Curves are graphical objects that can be manipulated, see the examples and the documentation of the parameters listed below for details.

Example 1

The most basic example of a curve in polar coordinates is a circle: Using a constant radius, the angle goes from to :

plot(plot::Polar([1, u], u = 0..2*PI))

MuPAD graphics

A constant angle, on the other hand, means a straight line through the origin:

plot(plot::Polar([r, 1], r = 0..1))

MuPAD graphics

plot::Polar accepts negative radii:

plot(plot::Polar([r, 1], r = -1..1))

MuPAD graphics

The most simple “interesting” example is probably Archimedes' spiral:

plot(plot::Polar([r, r], r = 0..5*PI))

MuPAD graphics

Example 2

Polar curves can be animated just like almost anything else:

plot(plot::Polar([r, a*r], r = 0..5*PI, a = -1..1))

MuPAD graphics

Example 3

In some cases, the default of evaluations on the curve is not sufficient and causes visible artifacts:

plot(plot::Polar([r, 4*r^2], r = 0..PI))

MuPAD graphics

One remedy for this problem is to increase the number of evaluation points:

plot(plot::Polar([r, 4*r^2], r = 0..PI, Mesh = 400))

MuPAD graphics

This method is, however, wasteful: Near the center, the initial density was perfectly sufficient, while on the outer edge still more points would be desirable. plot::Polar offers adaptive mesh refinement for exactly these situations. In the following example, we switch on adaptive mesh refinement with up to points introduced between each two consecutive points of the initial mesh: