Function Repository Resource:

# FivePointConic

Find a conic equation that passes through five given points

Contributed by: Ed Pegg Jr
 ResourceFunction["FivePointConic"][pts,{x,y}] returns the implicit Cartesian equation in the variables x and y of the conic section that goes through the points pts. ResourceFunction["FivePointConic"][pts] uses the formal variables x and y.

## Details

For random, uniformly-distributed points in a rectangle, the probability of a hyperbola is and the probability of an ellipse is .

## Examples

### Basic Examples (2)

Find a conic section through five points:

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Show the conic and points:

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### Scope (7)

Use formal variables:

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The results of a five point conic are usually a hyperbola:

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A random five point conic is also frequently an ellipse:

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A circle can also be the result of a five point conic:

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A parabola may also appear:

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Degenerate conics are also possible:

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See the properties of the non-degenerate conics with the resource function ConicProperties:

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### Neat Examples (5)

A function for characterizing a conic section:

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A sample conic section:

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The above polynomial defines a hyperbola:

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Degenerate conics appear frequently when selecting from a limited lattice:

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For random real-valued points, the result is always a hyperbola or ellipse:

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## Version History

• 1.0.0 – 22 July 2022