We have seen in the previous post the patterns made by the the tangents of certain parametric lines. This is a follow-up on that post and we will look at the same closed figures by plotting normals to them.

Definition: The simplest description of a normal is that it is a line segment perpendicular to the curve or the surface at the point where it is rooted. This is the Wikipedia link.

If you have a two-dimensional curve defined by the parametric equations x = x(t) and y = y(t)

Examples:

• Parametric Equation of a circle is \(x(t) = {Cos(t)}\) and \(y(t) = {Sin(t)}\)


• Tangent to any point on the circle \(T_{x}(t) = x'(t) = Cos(t)\) and \(T_{y}(t) = y'(t) = -Sin(t)\)


• Normal at any point on the circle would be \(N_{x}(t) = \frac{\partial^2 x(t)}{\partial^2 t} = -Sin(t)\) and \(N_{y}(t) = \frac{\partial^2 y(t)}{\partial^2 t} = -Cos(t)\)

Note: In this series, there will not be a separate image for the curve and the normals. The curve will be a closed line and it will be conspicuous enough to be visually separable from the other lines.

Three Lobes

Four Lobes

Five Lobes

The center of the above image zoomed in.

Six Lobes

Extras