Tangents

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] = (C1 + C2 Sin[n t]) Cos[t]
  • y[t] = (C1 + C2 Sin[n t]) Sin[t]
  • In this, the ratio of C1 and C2  along with ‘n’ determines the shape.

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A parametric function

Envelope[3]

Tangents drawn to that line

Tangent 3 Sample

 

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

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3 lobes

Envelope[3]

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

3 Lobes – 100 Tangents

Tang_3lob_lres100

3 Lobes – 400 Tangents

Tang_3lob_lres400

3 Lobes – 1000 Tangents

Tang_3lob_lres1000

3 Lobes – 5000 Tangents

Tang_3lob_lres5000


 

4 lobes

Envelope[4]

A square is formed by the envelop of these lines

Tangent 4

 

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5 Lobes

Envelope[5]

A pentagon is formed by the envelop of these lines

Tangent 5

 

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6 Lobes

Envelope[6]

A hexagon is formed by the envelop of these lines

Tangent 6

 

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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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One thought on “Tangents

  1. Pingback: Normals | Fermibot

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