Picking the ellipse
Two points plus two tangent directions are only four constraints, but an
ellipse has five degrees of freedom — so a one-parameter family
of ellipses satisfies your input. Each mode picks one member of that family a
different way.
- Roundest default
- Searches the family for the member closest to a circle (the smallest major / minor ratio). If the family contains a true circle, you get it. No value to set.
- Apex
- A slider that sweeps the whole family. It slides the near apex from the chord between the points (flat, at 0) out toward where the two tangent lines cross; 0.5 is exactly the parabola at the family’s edge, and everything in between is an ellipse.
- Rotation
- Finds the member whose axes sit at the angle you give (0–90° from the x-axis). Axis alignment pins down a single candidate; angles that don’t correspond to an ellipse raise an error.
- Aspect ratio
- Finds member(s) with the major / minor ratio you type (≥ 1). The condition is quadratic, so a ratio can match two ellipses — use “Next solution” to switch between them.
- Radius rx / ry
- Finds member(s) whose semi-major (rx) or semi-minor (ry) radius equals your value — up to two of them. rx can’t be shorter than half the distance between the points.
- Third point
- The single ellipse that also passes through a third point — drag the yellow point on the canvas, or type its coordinates.