Problem 58872. Find the Points Tangent to a Circle from an External Point

From a point where do the lines touch a circle tangentially?. The loldrup solution may provide some guidance and alternate method. I will elaborate a more reference frame modification geometric solution utilizing Matlab specific functions, rotation matrix, and translation matrix.
Given point(px,py) and circle [cx,cy,R] return the circle points [x3 y3;x4 y4] where lines thru the point are tangential to the circle. The line ([px,py],[x3,y3]) is tangential to circle [cx,cy,R] at circle point [x3,y3]. D>R.
The below figure is created based upon h=P=distance([cx,cy],[px,py])/2=D/2, translating (cx,cy) to (0,0), and rotating (px,py) to be on the Y-axis. From this manipulation two right triangles are apparent: [X,Y,R] and [X,h-Y,P]. Subtracting and simplifying these triangles leads to Y and two X values after substituting back into R^2=X^+Y^2 equation.
P^2=X^2+(h-Y)^2 and R^2=X^2+Y^2 after subtraction gives R^2-P^2=Y^2-(h-Y)^2 = Y^2-h^2+2hY-Y^2=2hY-h^2 thus
Y=(R^2-P^2+h^2)/(2h) =(R^2-P^2+P^2)/(2P)=R^2/(2P)=R^2/D
X=+/- (R^2-Y^2)^.5=+/- (R^2-(R^2/(2P))^2)^0.5=+/- R*(1-(R/D)^2)^0.5
The trick is to now un-rotate and translate this solution matrix using t=atan2(dx,dy), [cos(t) -sin(t);sin(t) cos(t)] and [cx cy]
This figure shows the point (px,py) rotated onto the Y-axis at position 2P. The circle (cx,cy) has been shifted to the origin with radius R. The green line shows a tangent at (x,y) from (px,py). A second tangent point is at (-x.y). D=2*P

Solution Stats

100.0% Correct | 0.0% Incorrect
Last Solution submitted on Dec 30, 2023

Solution Comments

Show comments

Problem Recent Solvers5

Suggested Problems

More from this Author295

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!