Protective geometry

[Guest]

How to construct a conic passing through four given points and tangent to a given conic?

[Guest]

To construct a conic passing through four given points and tangent to a given conic, you can follow these steps:

1. Identify the Given Points: First, identify the four given points through which the new conic must pass.
2. Formulate the Equation: Use the general equation of a conic section (which depends on the type of conic: ellipse, hyperbola, or parabola) to construct a system of equations. You'll have five unknown coefficients (or parameters) in your equation, and you can use the given points to set up a system of linear equations.
3. Solve the System: Solve the system of linear equations to find the coefficients of the new conic.
4. Form the New Conic: Once you have the coefficients, plug them into the general equation of the conic to form the equation of the new conic.
5. Tangent Condition: Use the given conic and its equation to ensure that the new conic is tangent to it. This will add an additional condition to your equations, usually resulting in one additional equation that you can solve along with the system formed by the given points.
6. Verify and Adjust: After finding the coefficients of the new conic, verify that it indeed passes through the given points and is tangent to the given conic. If adjustments are needed, refine the coefficients accordingly.
7. Draw the Conic: Once you have the equation of the new conic, you can plot it along with the given points and the given conic to visualize the result.

By following these steps, you can construct a conic passing through the specified points and tangent to the given conic.

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[Guest]

All conics form a one-parameter family through four fixed locations.

 F λ = F 1+ λF 2 = 0.

Tangency to a specific conic G=0 needs a double intersection, which can be discovered by removing the contact point from

F λ = 0, G = 0, F λ---G.

This results in a quadratic in λ, providing up to two conics through the four tangent points of  G.