Finding a Tangent to Circle for a Given Angle

[Cato]

I was hoping someone might be able to help me with tangent lines to circles. Please consider the figure below:



The challenge I have is to construct the "Target Tangent". The scenario is:

• I know what points A,B,C,D are and I already have the two blue tangents constructed
• I am then requested to draw a new tangent to the lower circle, for a given angle between the two blue tangents (in this case 20°)

I therefore am only given the angle of the target tangent, does anyone know how I can find the points E and F in order to draw it? Please note, I am only interested in the target tangent between the four known (blue) points, not the one that would appear on the other side of the circle.

Any help would be greatly appreciated.

N.B. The lines/angles I drew are rough approximations so please forgive any sloppiness.

 

[Dr.Peterson]

The challenge I have is to construct the "Target Tangent". The scenario is:

• I know what points A,B,C,D are and I already have the two blue tangents constructed
• I am then requested to draw a new tangent to the lower circle, for a given angle between the two blue tangents (in this case 20°)

I therefore am only given the angle of the target tangent, does anyone know how I can find the points E and F in order to draw it? Please note, I am only interested in the target tangent between the four known (blue) points, not the one that would appear on the other side of the circle.

Find the center of the lower circle; call it O. Then OE is a 20 degree rotation of OA; that gives you E. And the red tangent is the line through E perpendicular to OE. F is just the intersection of the tangent with the upper circle.

 

[Cato]

Find the center of the lower circle; call it O. Then OE is a 20 degree rotation of OA; that gives you E. And the red tangent is the line through E perpendicular to OE. F is just the intersection of the tangent with the upper circle.

Thank you Dr.P, you help me to solve a problem before so really appreciate your help again!

I have found the (x,y) co-ordinates of O. I am trying to do this in software, not by hand, so when you say OE is a 20° rotation of OA, which trig functions should I use to get the (x,y) co-ordinates of E? The same goes for F.

 

[Dr.Peterson]

when you say OE is a 20° rotation of OA, which trig functions should I use to get the (x,y) co-ordinates of E?

Subtract r cos(20) from the x coordinate of the center, and add r sin(20) to the y coordinate:

The same goes for F.

Find the equation of the tangent line and the equation of the top circle, and solve for the (lower) intersection.

Please show us an attempt, so we can have some idea of how much help you need. Do you know at least a little trig and algebra?

 

[Cato]

I managed to find point E of the rotated line OE using the following equations (where 'B' is 'E' in my case):



From there, getting the perpendicular "Target Tangent" vector simply involved inverting OE's components and negating one, i.e.:

TargetTangent.x = OE.y;
TargetTangent.y = -OE.x;

This seems to have produced exactly the right results. The key was recognising that OE is a 20° rotation of OA, and that the Target Tangent is the line perpendicular to OE. A really big thanks to you Dr.Peterson for pointing that out! Much appreciated.