How to find equation of a circle tangential to two straight lines?

Question

The image given above has two lines, AB and AC that are tangential to the circle with radius r. Points y1,y2 and the slope m of line AB are known. The graph represents a linear increase of speed (y axis) with respect to time (x axis). In-order to have a smooth transition of speed, edges of the line AB are curved using arc of a circle with radius r, as shown in the image. such that the circle is tangential to line AB and AC.

The unknowns values Width(to find the shift in time) and Height(to stop the curving) has to be found in-order to curve the edges of the line AB so that it appears like a spline. How can I find these values?

Answer 1

Hint:

Let $\angle BAC=\alpha $. Then the width $w $ and the height $h$ can be computed as: $$ w=r\tan\frac\alpha2=r\sqrt\frac{1-\cos\alpha}{1+\cos\alpha};\quad h=w\sin\alpha. $$

Answer 2

Let the slope of tangent be $ 2 \phi$ and let

$$ T= \frac{y_2-y_1}{x_2-x_1}=\tan 2 \phi,\text{find} \cos 2 \phi; $$

Solving quadratic equation for $t= \tan \phi$,

$$ T= \frac{2t}{1-t^2},\; t=\frac{\sqrt{1+T^2}-1}{T} $$

Line between the cutting point and circle center is bisector to the double right angled quadrilateral.

$$ W= r \; t, H= r (1- \cos 2 \phi).$$