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We look at the locus of the projection of the focus onto the normal, which turns about to be another parabola.
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We find the coordinates of the intersection of the normals to the parabola.
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We set the fixed point at the origin and the given line to be Y=b. The locus of the center of the circle is a parabola
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Set the short sides along the axes with lengths a and k-a. The equation is clearly a conic, and in fact a parabola:
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Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).
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A is at the origin, B on the line X=a. We take the envelope of the line through B at angle η to AB. The equation is complicated, but clearly a ...
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The focal chord of curvature is the chord of the circle of curvature which passes through the focus. Its length is simply expressed in terms of t.
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Here is the evolute, expressed in terms of the parameter (the parameter p, being four times the distance between vertex and focus).
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