In this two part series I though I would discuss some famous curves with interesting shapes.

### Cochleoid

Polar Form: *r* = *a* sin(*θ*)/*θ*

The name means the ‘snail-form’ curve. It was discussed by J Peck in 1700, however the form given above was discovered by Joseph Neuberg.

### Fermat’s Spiral

Polar Equation: *r*^{2} = *a*^{2}*θ*

This spiral was studied by Fermat in 1636. For any given positive value of *θ *there are two corresponding values of r: one negative and the other positive. Due to this, the spiral is symmetrical about the line y = -x.

### Lituus

Polar Equation: r^{2} = *a*^{2}/*θ*

The lituus curve, originated with Cotes in 1722, is the locus of the point P moving in such a way that the area of a circular sector remains constant.

### Watt’s Curve

Polar Equation: *r*^{2} = *b*^{2} – [*a* sin(*θ*) ± √(*c*^{2} – *a*^{2}cos^{2}(*θ*))]^{2}

This curve was named after James Watt, a Scottish engineer who developed the steam engine, due to the fact that the curve comes from the linkages of rods connecting two wheels of equal diameter.

### Newton’s Diverging Parabolas

Cartesian Equation: *ay*^{2} = *x*(*x*^{2} – 2*bx* + *c*), *a* > 0

The above equation represents the third class of Newton’s classification of cubic curves. These cases can be divided as follows:

**All the roots are real and unequal.** The figure is a diverging parabola with the shape of a bell, with an oval at its vertex. *This is the case depicted in the image.*

**Two of the roots are equal.** A parabola will be formed, either Nodated (by touching an oval) or Punctate (by having the oval infinitely small).

**The three roots are equal.** This is the semicubical parabola.

**Only one real root.** If two of the roots are not real, there will be a pure parabola with a bell-like form.

Stay tuned for part 2 on Friday! M x

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