Find an expression for the signature of each of the following boundaries, and plot the signatures.

i) An equilateral triangle

ii) A rectangle

iii) An ellipse

To find the signatures for the boundaries of these shapes, we need to first define what a "signature" means in the context of image processing. The signature of a boundary is typically a function of the distance from a reference point along the boundary, as the boundary is traced.

### i) Equilateral Triangle
For an equilateral triangle, let the vertices be at points \( A(0, h) \), \( B(-b, 0) \), and \( C(b, 0) \), where \( h \) is the height of the triangle and \( b \) is half of its base.

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