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Narasimham
Narasimham
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We find new configurations by extensions, not by scaling.

By enlarging first circle radius by an arbitrary amount $\delta_1$ and contracting the second circle radius by $\delta_2$ to maintain tangential contact, a set of centers and their radii of the two circles can always be found. There is no unique result.

To find $\delta_2$ as a function of $\delta_1$ the sum of vector projections ( found by vector dot products) of two radii, and projection of line of centers is always constant equal to distance between given two fixed points of contacting circles.

Before applying the $\delta s$

$$ r_1 \cos \alpha + (r_1+r_2) \cos \gamma + r_2 \cos \beta = d. $$

enter image description here

By enlarging first circle radius by an arbitrary amount $\delta_1$ and contracting the second circle radius by $\delta_2$ to maintain tangential contact, a set of centers and their radii of the two circles can always be found. There is no unique result.

To find $\delta_2$ as a function of $\delta_1$ the sum of vector projections ( found by vector dot products) of two radii, and projection of line of centers is always constant equal to distance between given two fixed points of contacting circles.

Before applying the $\delta s$

$$ r_1 \cos \alpha + (r_1+r_2) \cos \gamma + r_2 \cos \beta = d. $$

enter image description here

We find new configurations by extensions, not by scaling.

By enlarging first circle radius by an arbitrary amount $\delta_1$ and contracting the second circle radius by $\delta_2$ to maintain tangential contact, a set of centers and their radii of the two circles can always be found. There is no unique result.

To find $\delta_2$ as a function of $\delta_1$ the sum of vector projections ( found by vector dot products) of two radii, and projection of line of centers is always constant equal to distance between given two fixed points of contacting circles.

Before applying the $\delta s$

$$ r_1 \cos \alpha + (r_1+r_2) \cos \gamma + r_2 \cos \beta = d. $$

enter image description here

added 117 characters in body
Source Link
Narasimham
Narasimham
  • 43.2k
  • 7
  • 48
  • 119

By enlarging first circle radius by an arbitrary amount $\delta_1$ and contracting the second circle radius by $\delta_2$ to maintain tangential contact, a set of centers and their radii of the two circles can always be found. There is no unique result.

To find $\delta_2$ as a function of $\delta_1$ the sum of vector projections ( found by vector dot products) of two radii,and and projection of line of centers is always constant equal to distance between given two fixed points of contacting circles.

Before applying the $\delta s$

$$ r_1 \cos \alpha + (r_1+r_2) \cos \gamma + r_2 \cos \beta = d. $$

enter image description here

By enlarging first circle by an arbitrary amount $\delta_1$ and contracting the second circle by $\delta_2$ to maintain tangential contact, a set of centers and their radii of the two circles can always be found. There is no unique result.

To find $\delta_2$ as a function of $\delta_1$ the sum of vector projections of two radii,and projection of line of centers is always constant equal to distance between given two fixed points of contacting circles.

enter image description here

By enlarging first circle radius by an arbitrary amount $\delta_1$ and contracting the second circle radius by $\delta_2$ to maintain tangential contact, a set of centers and their radii of the two circles can always be found. There is no unique result.

To find $\delta_2$ as a function of $\delta_1$ the sum of vector projections ( found by vector dot products) of two radii, and projection of line of centers is always constant equal to distance between given two fixed points of contacting circles.

Before applying the $\delta s$

$$ r_1 \cos \alpha + (r_1+r_2) \cos \gamma + r_2 \cos \beta = d. $$

enter image description here

Source Link
Narasimham
Narasimham
  • 43.2k
  • 7
  • 48
  • 119

By enlarging first circle by an arbitrary amount $\delta_1$ and contracting the second circle by $\delta_2$ to maintain tangential contact, a set of centers and their radii of the two circles can always be found. There is no unique result.

To find $\delta_2$ as a function of $\delta_1$ the sum of vector projections of two radii,and projection of line of centers is always constant equal to distance between given two fixed points of contacting circles.

enter image description here