Curves

Cartesian Oval

Cartesian equation:: $((1 - m^{2}) (x^{2} + y^{2}) + 2 m^{2} c x + a^{2} - m^{2} c^{2})^{2} = 4 a^{2} (x^{2} + y^{2}) ((1 - m^{2})(x^{2} + y^{2}) + 2m^{2}cx + a^{2} - m^{2}c^{2})^{2} = 4a^{2}(x^{2} + y^{2})$

Description

This curve

C C

consists of two ovals so it should really be calledCartesian Ovals. It is the locus of a point

P P

whose distances

s s

and

t t

from two fixed points

S S

and

T T

satisfy

s + m t = a s + mt = a

. When

c c

is the distance between

S S

and

T T

then the curve can be expressed in the form given above.

The curves were first studied byDescartes in1637 and are sometimes called the 'Ovals ofDescartes'.

The curve was also studied byNewton in his classification of cubic curves.

The Cartesian Oval has bipolar equation

r + m r^{'} = a r + mr' =a

.

If

m = \pm 1 m = ±1

then the Cartesian Oval

C C

is a central conic while if

m = a / c m = a/c

then the curve is aLimacon of Pascal(Étienne Pascal). In this case the inside oval touches the outside one.

Cartesian Ovals are anallagmatic curves.

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Curves

Cartesian Oval

Description

Associated Curves