Polar Functions

Some noteworthy polar functions that can be randomly generated using this applet include:

Conic Sections: A conic section is a shape created by intersecting a cone and a plane. Four types of graphs can emerge: a circle, an ellipse, a parabola, or a hyperbola. A circle has the simplest polar form, \[ r = a\sin(t) \qquad \mbox{ or } \qquad r = a\cos(t) \] The diameter of the circle will be \(|a|\). The center of the circle will be on the horizontal axis if the function includes \(\cos(t)\), and on the vertical axis if the function includes \(\sin(t)\).
The other conic sections can be described by a function of the form \[ r = \frac{a}{1\pm b\cos(t)} \] If \(0 < b < 1\), the conic will be an ellipse. If \(b=1\), we obtain a parabola, and if \(b > 1\), we'll have a hyperbola.
Cardioids and Limaçons: A limaçons (French for "snail") is a graph whose polar function has the form \[ r = a \pm b\sin(t) \qquad \mbox{ or } \qquad r = a \pm b\cos(t) \] where \(a \neq b\) and both \(a>0\) and \(b>0\). If \(a \lt b \), then we'll have an "inner limaçon" with a loop inside. If \(a=b\), we obtain a cardioid, named for its resemblance to a heart.
Limaçons: A limaçons (French for "snail") is a graph whose polar function has the form \[ r = a \pm b\sin(t) \qquad \mbox{ or } \qquad r = a \pm b\cos(t) \] where \(a \neq b\) and both \(a>0\) and \(b>0\). If \(a \lt b \), then we'll have an "inner limaçon" with a loop inside.
Spirals: An Archimedean spiral has polar form \(r = a+bt\), where \(a\) determines the center of the circle and \(b\) determines how the rate of growth of the spiral. These are categorized by the equal spacing between "arms" of the spiral, and are defined as the path left by a point moving at a constant velocity along a line moving with constant angular velocity.
Roses: A rose curve, named for its resemblance to a flower, is a graph whose polar function has the form \[ r = a\sin(nt) \qquad \mbox{ or } \qquad r = a\cos(nt) \] The number of "petals" depends on \(n\). If \(n\) is odd, the rose will have \(n\) petals. If \(n\) is even, the rose will have \(2n\) petals. The size of the rose is determined by \(a\). Adding a value less than \(a\) to the function results in a double petal.
These roses are just one example of rhodonea curves, in which \(n\) can be a rational value.

This applet illustrates the effects of applying various linear transformations to objects in \( \mathbb{R}^2 \). Choose a shape to apply transformations to, and zoom and in out using the slider. The \(2 \times 2\) transformation matrix can be entered directly, or you can choose one of the preset transformations listed. Choosing a preset transformation will update the transformation matrix automatically.

Check the "Combine Transformation" box to compose transformations. Transformations are composed by multiplying on the left by subsequent matrices. For instance, suppose \(R\) is a rotation matrix, and \(S\) is a matrix that scales the object. Then \(RS\) is a transformation that first scales the object, and then rotates it, while \(SR\) is a transformation that rotates the object, followed by scaling. To enter \(RS\), check the "Combine Transformation" box, then choose an amount to scale followed by the rotation. \(SR\) is entered in the opposite order - first rotate, then scale. Note that these transformations will generally not be equal, since matrix multiplication is not commutative.

An additional feature of the applet is the ability to see where each point \((x,y)\) of the object is sent by a transformation \(T\). Clicking and holding the mouse while moving over any point will reveal a pair of vectors \((x,y)\) and \(T(x,y)\). If these vectors are scalar multiples (they overlap), the vector is an eigenvector of the transformation.

If the orientation of the shape has been reversed due to a reflection, the color of the shape will change from blue to grey. The original position of the shape is displayed as a dotted grey outline.