Example 7- Two Quadratic Splines

Example 7- Two Quadratic Splines

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In the diagram below, D and E are located at proportion t along AB and BC respectively.  F is the intersection of AE and CD.  G is the point proportion t along the line DE.  We examine the loci of F and G as t varies from 0 to 1. 

Observing the parametric form of the curves we see that one is a parametric quadratic, while the other is a rational quadratic.  Implicit forms are both conics (and almost, but not quite, identical).

 

What types of conics are they?  Extending the curves a little can give a clue:

 

The upper curve looks like a parabola, the lower certainly does not.

Can you show this is generally true from the algebraic equations of the curves?