A Solution to the Problem of Apollonius Using Vector Dot Products
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The document presents a solution to the Problem of Apollonius, which is to construct circles tangent to three given coplanar circles, using only vector dot products. It defines variables to represent the circles and derives equations that lead to an expression for the radius of the solution circle that encloses none of the three given circles. The solution method finds the points of tangency of two tangent circles, one enclosing and one not enclosing the three initial circles.
![A Solution to the Problem of Apollonius Using
Vector Dot Products
Jim Smith
QueLaMateNoTeMate.webs.com
email: nitac14b@yahoo.com
LinkedIn Group: Pre-University Geometric Algebra
October 6, 2016
Abstract
To the collections of problems solved via Geometric Algebra (GA) in
[1]-[11], this document adds a solution, using only dot products, to the
Problem of Apollonius. The solution is provided for completeness and for
contrast with the GA solutions presented in [4].
The Problem of Apollonius: Given three coplanar circles, construct
the circles that are tangent to all three of them, simultaneously.
1](https://image.slidesharecdn.com/apolloniusdotprodsonly-161007004057/85/A-Solution-to-the-Problem-of-Apollonius-Using-Vector-Dot-Products-1-320.jpg)
![The geometric interpretation of Eq. (3) is shown in Fig. 3: Our solution
method has found the points of tangency of two tangent circles. One of them
encloses all three of the givens, while the other encloses none of them.
Figure 3: The geometric interpretation of Eq. (3): Our solution method has
found the points of tangency of two tangent circles. One of them encloses all
three of the givens, while the other encloses none of them.
References
[1] J. Smith, 2015, “From two dot products, determine
an unknown vector using Geometric (Clifford) Algebra”,
https://www.youtube.com/watch?v=2cqDVtHcCoE .
[2] J. Smith, 2016, “Rotations of Vectors Via Geometric Alge-
bra: Explanation, and Usage in Solving Classic Geometric ‘Con-
struction’ Problems” (Version of 11 February 2016). Available at
http://vixra.org/abs/1605.0232 .
[3] J. Smith, 2016,“Solution of the Special Case ‘CLP’ of the Problem of
Apollonius via Vector Rotations using Geometric Algebra”. Available at
http://vixra.org/abs/1605.0314.
[4] J. Smith, 2016,“The Problem of Apollonius as an Opportunity for
Teaching Students to Use Reflections and Rotations to Solve Ge-
ometry Problems via Geometric (Clifford) Algebra”. Available at
http://vixra.org/abs/1605.0233.
4](https://image.slidesharecdn.com/apolloniusdotprodsonly-161007004057/85/A-Solution-to-the-Problem-of-Apollonius-Using-Vector-Dot-Products-4-320.jpg)
![[5] J. Smith, 2016,“A Very Brief Introduction to Reflections in 2D Geometric
Algebra, and their Use in Solving ‘Construction’ Problems”. Available at
http://vixra.org/abs/1606.0253.
[6] J. Smith, 2016,“Three Solutions of the LLP Limiting Case of the Problem
of Apollonius via Geometric Algebra, Using Reflections and Rotations”.
Available at http://vixra.org/abs/1607.0166.
[7] J. Smith, 2016,“Simplified Solutions of the CLP and CCP Limiting Cases
of the Problem of Apollonius via Vector Rotations using Geometric Alge-
bra”. Available at http://vixra.org/abs/1608.0217.
[8] J. Smith, 2016, “Additional Solutions of the Limiting Case ‘CLP’ of the
Problem of Apollonius via Vector Rotations using Geometric Algebra”.
Available at http://vixra.org/abs/1608.0328.
[9] J. Smith, 2016, “Some Solution Strategies for Equations that Arise in Ge-
ometric (Clifford) Algebra”. Available at http://vixra.org/abs/1610.0054.
[10] J. Smith, “Geometric Algebra of Clifford, Grassman, and Hestenes”,
https://www.youtube.com/playlist?list=PL4P20REbUHvwZtd1tpuHkziU9rfgY2xOu.
[11] J. Smith, “Geometric Algebra (Clifford Algebra)”,
https://www.geogebra.org/m/qzDtMW2q#chapter/0.
5](https://image.slidesharecdn.com/apolloniusdotprodsonly-161007004057/85/A-Solution-to-the-Problem-of-Apollonius-Using-Vector-Dot-Products-5-320.jpg)
A Solution to the Problem of Apollonius Using Vector Dot Products
- 1. A Solution to the Problem of Apollonius Using Vector Dot Products Jim Smith QueLaMateNoTeMate.webs.com email: nitac14b@yahoo.com LinkedIn Group: Pre-University Geometric Algebra October 6, 2016 Abstract To the collections of problems solved via Geometric Algebra (GA) in [1]-[11], this document adds a solution, using only dot products, to the Problem of Apollonius. The solution is provided for completeness and for contrast with the GA solutions presented in [4]. The Problem of Apollonius: Given three coplanar circles, construct the circles that are tangent to all three of them, simultaneously. 1
- 2. 1 Introduction The Problem of Apollonius reads, “Given three coplanar circles, construct the circles that are tangent to all three of them, simultaneously.” Figure 1: The Problem of Apollonius: “Given three coplanar circles (e.g. C1, C2, and C3), construct the circles that are tangent to all three of them, simulta- neously.” There are eight solution circles, two of which (C4 and C5) are shown here. Because this problem, along with several of its limiting cases, has been treated at length in the references, the solution presented here will leave the details to the reader. We will identify only the solution circle that encloses none of the givens. (I.e., C4 in Fig. 1.) 2 Solution 2.1 Defining Variables that are Amenable to Treatment via GA The variables that we will use are shown in Fig. 2. 2.2 Solution Strategy Our strategy will be to express the vector c4 in two ways, from which we will then derive expressions for ˆt and r4. 2
- 3. Figure 2: Definition of variables used for identifying the solution circle C4. 2.3 Formulating and Solving the Equations We begin by equating two expressions for the vector c4 c4 = (r1 + r4)ˆt = c2 + (r2 + r4) ˆw2 ∴ (r1 + r4)ˆt − c2 = (r1 + r4) ˆw2. Now, we’ll square both sides to eliminate the unknown vector ˆw2, after which we’ll solve for r4: r4 = c2 2 + r1 2 − r1 2 − 2r1c2 · ˆt 2 r2 − r1 + c2 · ˆt . (1) Similarly, c4 = (r1 + r4)ˆt = c3 + (r3 + r4) ˆw3; (r1 + r4)ˆt − c3 = (r3 + r4) ˆw3; and r4 = c3 2 + r3 2 − r1 2 − 2r1c3 · ˆt 2 r3 − r1 + c3 · ˆt . (2) Equating the expressions for r4 from Eqs. (1) and (2), c2 2 + r1 2 − r1 2 − 2r1c2 · ˆt r2 − r1 + c2 · ˆt = c3 2 + r3 2 − r1 2 − 2r1c3 · ˆt r3 − r1 + c3 · ˆt . After cross-multiplying, simplifying, and rearranging as explained in the references, we obtain c3 2 − (r3 − r1) 2 c2 − c2 2 − (r2 − r1) 2 c3 W e ll call this vector “z”. ·ˆt = (r3 − r1) c2 2 − r2 2 − (r2 − r1) c3 2 − r3 2 + (r3 − r2) 2 r1 2 . (3) 3
- 4. The geometric interpretation of Eq. (3) is shown in Fig. 3: Our solution method has found the points of tangency of two tangent circles. One of them encloses all three of the givens, while the other encloses none of them. Figure 3: The geometric interpretation of Eq. (3): Our solution method has found the points of tangency of two tangent circles. One of them encloses all three of the givens, while the other encloses none of them. References [1] J. Smith, 2015, “From two dot products, determine an unknown vector using Geometric (Clifford) Algebra”, https://www.youtube.com/watch?v=2cqDVtHcCoE . [2] J. Smith, 2016, “Rotations of Vectors Via Geometric Alge- bra: Explanation, and Usage in Solving Classic Geometric ‘Con- struction’ Problems” (Version of 11 February 2016). Available at http://vixra.org/abs/1605.0232 . [3] J. Smith, 2016,“Solution of the Special Case ‘CLP’ of the Problem of Apollonius via Vector Rotations using Geometric Algebra”. Available at http://vixra.org/abs/1605.0314. [4] J. Smith, 2016,“The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Ge- ometry Problems via Geometric (Clifford) Algebra”. Available at http://vixra.org/abs/1605.0233. 4
- 5. [5] J. Smith, 2016,“A Very Brief Introduction to Reflections in 2D Geometric Algebra, and their Use in Solving ‘Construction’ Problems”. Available at http://vixra.org/abs/1606.0253. [6] J. Smith, 2016,“Three Solutions of the LLP Limiting Case of the Problem of Apollonius via Geometric Algebra, Using Reflections and Rotations”. Available at http://vixra.org/abs/1607.0166. [7] J. Smith, 2016,“Simplified Solutions of the CLP and CCP Limiting Cases of the Problem of Apollonius via Vector Rotations using Geometric Alge- bra”. Available at http://vixra.org/abs/1608.0217. [8] J. Smith, 2016, “Additional Solutions of the Limiting Case ‘CLP’ of the Problem of Apollonius via Vector Rotations using Geometric Algebra”. Available at http://vixra.org/abs/1608.0328. [9] J. Smith, 2016, “Some Solution Strategies for Equations that Arise in Ge- ometric (Clifford) Algebra”. Available at http://vixra.org/abs/1610.0054. [10] J. Smith, “Geometric Algebra of Clifford, Grassman, and Hestenes”, https://www.youtube.com/playlist?list=PL4P20REbUHvwZtd1tpuHkziU9rfgY2xOu. [11] J. Smith, “Geometric Algebra (Clifford Algebra)”, https://www.geogebra.org/m/qzDtMW2q#chapter/0. 5