Where is Point proportional along curve for conics?

Where is Point proportional along curve for conics?

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The best way to understand the location of Point proportional along curve command for conics is to see how we construct it geometrically for each conic:

Ellipse

The ellipse with foci A and B is inscribed in circle, center M.  Draw the radius MN at angle t to the major axis and drop the segment NO perpendicular to the major axis of the ellipse.  When the intersection of NO with the ellipse (point C) is constrained to be t proportional along the ellipse, it's coordinates will be (a cos(t), b sin(t)).

point pro_ellipse 

Parabola

C lies on the parabola and BC is perpendicular to the axis AB of the parabola. Point D is located proportion t along the segment. Point F is the intersection of the perpendicular to BC through D with the parabola.  It has the coordinates (2at, at2) when it is constrained to parametric location t on this parabola.

point pro_parab 

Hyperbola

CD is the perpendicular projection of C onto the axis of the hyperbola, GF is the circle centered at the center of the hyperbola which goes through the intersections of the hyperbola with its axis.  H is the point of contact of this circle with the tangent from D.  We can see that the angle DGH is the same as the parameter value.  When point C is constrained to be at parametric location t along the curve, its coordinates are (a/cos(t), b sin(t)/cos(t)) on this hyperbola.

point pro_hyperb