The chain of six spheres can be rotated about the central sphere without affecting their tangencies, showing that there is an infinite family of solutions for this case.
42.
An intersection of these two hyperbolas ( if any ) gives the center of a solution circle that has the chosen internal and external tangencies to the three given circles.
43.
In 3D, the animation curves of a blocked shot are often created using " stepped " or " square " tangencies, which provides no interpolation between animation poses.
44.
The two " tangency chords " of a tangential quadrilateral are the line segments that connect points on opposite sides where the incircle is tangent to these sides.
45.
This follows because any solution circle would have to cross over the middle circle to move from its tangency to the inner circle to its tangency with the outer circle.
46.
This follows because any solution circle would have to cross over the middle circle to move from its tangency to the inner circle to its tangency with the outer circle.
47.
The full set of solutions to Apollonius'problem can be found by considering all possible combinations of internal and external tangency of the solution circle to the three given circles.
48.
Thus, as the solution circle swells, the internally tangent given circles must swell in tandem, whereas the externally tangent given circles must shrink, to maintain their tangencies.
49.
Equivalently, if we zoom in more and more at the point of tangency the manifold appears to become more and more straight, asymptotically tending to approach the tangent space.
50.
Specifically, the point of tangency between any isoquant and an isocost line gives the lowest-cost combination of inputs that can produce the level of output associated with that isoquant.
