Draw an arc given two points and a radius

Question

Draw an arc given two points and a radius.

How can I understand the answer given here?

Draw an Arc from a start point, end point and a radius

https://stackoverflow.com/a/36872927/261439

Answer 1

Let start point is S, end point is E

Center C lies at middle perpendicular to SE segment, so we should find vectors:

se = (E.x - S.x, E.y - S.y)
perp = (-se.y, se.x) = (S.y - E.y, E.x - S.x)
perp_normalized = perp / Len(perp)
L = Len(perp) = sqrt((S.y - E.y)^2+(S.x - E.x)^2)
perp_normalized = (S.y - E.y)/L, (E.x - S.x)/L

and middle point of SE segment

M = ((E.x + S.x)/2, (E.y + S.y)/2)

Distance from M to circle center is

D = sqrt(R^2-L^2/4) 

Now we can express circle center C coordinates using parametric equation, there are two possible variants:

C = M +/- perp_normalized*D

C1.x = M.x + perp_normalized.x * D
C1.y = M.y + perp_normalized.y * D

C2.x = M.x - perp_normalized.x * D
C2.y = M.y - perp_normalized.y * D

Having center, you can define parameters needed for drawing of arc in your framework

Answer 2

C is the midpoint of DE, so AB and DE are perpendicular (midpoint theorem)

AC=sqrt(AD^2-CD^2) (Pythagoras theorem)

DE=sqrt(AD^2-CD^2) (Pythagoras theorem)

Triangle ACF and EDG are similar triangles, scaled by ratio AC:DE=AF:EG=CF:DG

DG and EG are known. AC and DE are given above. Then we get CF and AF. (And it can be proven that dX(G->D) and dY(C->F) same sign; dX(E->G) and dY(F->A) revsrse sign.)

Therefore we get the center of circle.

And we can get points on the arc by rotating D around A. (rotation formula)