For a continuously differentiable line, a tangent can be drawn to it at any point. Look the following pair for example. The first one is a line and the second line shows the same diagram with some tangents drawn to it. Let us look at some equations first.

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The equations

A figure with ‘n’ lobes can be created by the following parametric equation

- x[t] = (C + Sin[n t]) Cos[t]
- y[t] = (C + Sin[n t]) Sin[t]

Where

- the constant ‘C’ determines the size of the structure. It is basically the radius of the circle upon which you are adding the Sin[n t].
- n is the constant that determines the number of lobes. For obtaining three lobes, use ‘3’ for n.

More complicated equation would read as

- x[t] = (C
_{1}+ C_{2}Sin[n t]) Cos[t] - y[t] = (C
_{1}+ C_{2}Sin[n t]) Sin[t] - In this, the ratio of C
_{1}and C_{2}along with ‘n’ determines the shape.

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A parametric function

Tangents drawn to that line

The process can be done more rigorously to obtain some interesting results.

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#### 3 lobes

Do note that the envelope of the lines obtained is similar to the line itself

#### 3 Lobes – 100 Tangents

#### 3 Lobes – 400 Tangents

#### 3 Lobes – 1000 Tangents

#### 3 Lobes – 5000 Tangents

4 lobes

A square is formed by the envelop of these lines

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#### 5 Lobes

A pentagon is formed by the envelop of these lines

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#### 6 Lobes

A hexagon is formed by the envelop of these lines

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A regular polygon of “n” sides can be created by creating a line with the same number of lobes.

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A transformation from the various lobes can be seen in the videos below.

#### Transformation 3 to 4

Please watch in Fullscreen mode for best quality.

#### Transformation 4 to 5

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