I'm trying my best to figure a circle problem out!
Consider a circle, or an ellipse (a circle scaled either on the local X or Y axis). For any two points in quadrant 1, what is the equation of the circle/ellipse such that it passes through both points, having those points rest on quadrant 1 of the circle/ellipse if it was a unit circle, or in other words, have it rest on the arc from $0$ to $\pi/2$. I figure there's likely an infinite number of solutions depending on the scale, and that's desired, because I'm looking to have input parameters:
point1X, point1Y, point2X, point2Y, circleScaleX, circleScaleY. It's possible that there's still more than one solution with those inputs, and if so, what else do I have to constrain?
The point of what I'm trying to accomplish isn't really to get one solution anyway, but to get a variable shape arc between the two points, being able to tweak it until it looks good. So even if there's a third circle variable of some kind to set to result in the one solution I need to paint the arc to the screen, that's cool with me.
Now that I think about it, I think a constraint for the points is that
point1X < point2X and
point1Y > point2Y so that we always result in the arc "roundness" pointing out in the positive direction of $y = x$.
I should note that it's a requirement that the arc is composed of a segment of a circle or ellipse, I can't use something like Bézier curves.
Thanks ahead of time, all the best!