Curves #2

This is the second post in my two part series on various, famous curves. Click here to check out part 1!

Hypocycloid

Parametric Cartesian Equation: x = (ab) cos(t) + b cos((a/b – 1)t) // y = (ab) sin(t) – b sin((a/b – 1)t)

A hypocycloid is the curve traced by a point on the circumference of a circle that is rolling on the interior of another circle.

Source: Wikipedia

Talbot’s Curve

Parametric Cartesian Equation: x = (a2 + f2sin2(t))cos(t)/a // y = (a2 – 2f2 + f2sin2(t))sin(t)/b

Investigated by Talbot, this curve is the negative pedal of an ellipse with respect to its centre. It has four cusps and two nodes provided that the eccentricity of the ellipse squared > o.5.

Conchoid

Cartesian Equation: (xb)2(x2 + y2) – a2x2 = 0

Polar Equation: r = a + b sec(θ)

The conchoid was studied by Greek mathematician Nicomedes (about 200BC). It can be used to solve the problems of the duplication of a cube and trisecting an angle. In fact, Newton said it should be a ‘standard’ curve.

The conchoid was used in the construction of ancient buildings; the vertical section of columns was made in the shape of the loop of the conchoid.

Devil’s Curve

Cartesian Equation: y4x4 + a y2 + b x2 = 0

The Devil’s curve was studied both by Gabriel Cramer in 1750 and Lacrois in 1810.

Dürer’s Shell Curves


Cartesian Equation: (x2 + xy + axb2)2 = (b2x2)(xy + a)2

These curves appear in ‘Instruction in measurement with compasses and straight edge‘, published by Dürer in 1525. This curve arose from his work on perspective.

“He constructed the curve in the following way. He drew lines QRP and P’QR of length 16 units through Q (q, 0) and R (0, r) where q + r = 13. The locus of P and P’ as Q and R move on the axes is the curve. Dürer only found one of the two branches of the curve.”

Epicycloid

Parametic Cartersian Equation: x = (a + b) cos(t) – b cos((a/b + 1)t) // y = (a + b) sin(t) – b sin((a/b + 1)t)

An epicycloid is a curve produced by tracing the path of a chosen point of a circle which rolls without slipping around a fixed circle.

What’s your favourite curve? M x

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