Descartes Oval

Descartes Oval - a fourth-order plane algebraic curve representing the geometrical location of points for which the sum of the distances $r_{1}$ ${\ displaystyle r_ {1}}$ $r_ {1}$ and $r_{2}$ ${\ displaystyle r_ {2}}$ $r_ {2}$ up to two points $F_{1}$ ${\ displaystyle F_ {1}}$ $F_ {1}$ and $F_{2}$ ${\ displaystyle F_ {2}}$ $F_ {2}$ called tricks multiplied by constants $p_{1}$ ${\ displaystyle p_ {1}}$ $p_ {1}$ and $p_{2}$ ${\ displaystyle p_ {2}}$ $p_ {2}$ is constant, that is:

p_{1}r_{1}+p_{2}r_{2}=d.

{\ displaystyle p_ {1} r_ {1} + p_ {2} r_ {2} = d.}

{\ displaystyle p_ {1} r_ {1} + p_ {2} r_ {2} = d.}

Curve Equation

This curve is described by the equation:

(x^{2}+y^{2}-2ax)^{2}=b^{2}(x^{2}+y^{2})+c\;,

{\ displaystyle (x ^ {2} + y ^ {2} -2ax) ^ {2} = b ^ {2} (x ^ {2} + y ^ {2}) + c \ ;,}

(x^{2}+y^{2}-2ax)^{2}=b^{2}(x^{2}+y^{2})+c\;,

where a , b and c are the constants associated with the parameters p ₁ , p ₂ and d .

At $c=0$ ${\ displaystyle c = 0}$ $c=0$ Descartes' oval is a snail of Pascal .

If a $p_{1}=p_{2},$ ${\ displaystyle p_ {1} = p_ {2},}$ $p_{1}=p_{2},$ then Descartes' oval is an ellipse , in the case $p_{1}=-p_{2},$ ${\ displaystyle p_ {1} = - p_ {2},}$ $p_{1}=-p_{2},$ - hyperbole .

This curve was first studied and described by Rene Descartes in 1637. Descartes constructed these ovals when solving the optics problem: he was looking for a curve that would refract the rays coming from one point, so that the refracted rays would pass through another given point.

Descartes Oval Examples

a = 1, b = 1, c = 0

a = 1, b = 1, c = 1

a = 1, b = 1, c = -1

a = 1, b = 1, c = 0.05

a = 1.5, b = 0, c = 0.5

Links

D.K. Bobylev . Cartesian ovals // Brockhaus and Efron Encyclopedic Dictionary : in 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.
Ocartes of Descartes

Descartes Oval

Curve Equation

Descartes Oval Examples

See also

Links

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