An additional brief solution of the CPP limiting case of the Problem of Apollonius via Geometric Algebra (GA)
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This document presents a geometric algebra approach to solving the CPP limiting case of the problem of Apollonius, which involves constructing circles tangent to a given circle and passing through two external points. It derives expressions for the radii of the circles and shows how these relate to the geometric elements identified in previous works. The solution indicates that two solution circles exist, with points of tangency that are reflections of each other with respect to a specific vector.
![An additional brief solution of the CPP limiting
case of the Problem of Apollonius via Geometric
Algebra (GA)
Jim Smith
QueLaMateNoTeMate.webs.com
email: nitac14b@yahoo.com
September 25, 2016
Abstract
This document adds to the collection of GA solutions to plane-geometry
problems, most of them dealing with tangency, that are presented in [1]-
[7]. Reference [1] presented several ways of solving the CPP limiting case
of the Problem of Apollonius. Here, we use ideas from [6] to solve that
case in yet another way.
1 Statement of the problem
The CPP limiting case of the Problem of Apollonius reads,
“Given a circle and two points outside of it, construct the circles
that are tangent to the given one, and that also pass through both of
the given points” (Fig. 1).
2 Expressing elements of the problem in ways
that facilitate solution via GA
We should note that instead of
the angles shown, we could have
used angles of rotation from
b − t to ˆt, and from b − c2 to ˆt.
Experience gained from earlier work (especially [6]) suggests that we use
the elements identified in Fig. 2. Note, especially, that the angles θ and 2θ
terminate at the line that connects the centers of the two circles, and have as
their vertices the point of tangency and the center of the solution circle.
1](https://image.slidesharecdn.com/apolloniuscppviar2-160925231827/85/An-additional-brief-solution-of-the-CPP-limiting-case-of-the-Problem-of-Apollonius-via-Geometric-Algebra-GA-1-320.jpg)
![Figure 1: The CPP limiting case of the Problem of Apollonius: Given a circle
and two points outside of it, construct the circles that are tangent to the given
one, and that also pass through both of the given points.
Figure 2: The point of tangency t, and the geometric elements of the problem
that will be used to identify t via GA.
3 Solution
We will follow [6] in beginning by deriving an expression for r2 and r1+r2, which
we will then use in an equation that equates two expressions for the multivector
e2θi
. Please see [6] for further details.
3.1 Expressions for r2 and r1 + r2
From Fig. 2, we can derive that
(r1 + r2)ˆt =
a + b
2
+
(a − b) i
a − b
r2
2 −
a − b
2
2
.
We’ll rearrange that as
(r1 + r2)ˆt −
a + b
2
=
(a − b) i
a − b
r2
2 −
a − b
2
2
,
2](https://image.slidesharecdn.com/apolloniuscppviar2-160925231827/85/An-additional-brief-solution-of-the-CPP-limiting-case-of-the-Problem-of-Apollonius-via-Geometric-Algebra-GA-2-320.jpg)
![then square both sides. After simplifying and solving for r2, we find that
r2 =
2r1 (a + b) · ˆt − a2
− b2
− 2r1
2
4r1 − 2 (a + b) · ˆt
. (1)
From that result, we can then derive
r1 + r2 =
2r1
2
− a2
− b2
4r1 − 2 (a + b) · ˆt
. (2)
3.2 Equating two expressions for the multivector e2θi
Note that of the two given
points, only one of them (in this
case, a) figures in multivector
that we are using. We make no
use of the other point, than to
develop an expression for r2 in
Eq. (3.1).
Still following [6], we see from Fig. 2 that
a − t
a − t
ˆt
=eθi
a − t
a − t
ˆt
=eθi
=
a − c2
a − c2
ˆt
=e2θi
,
from which
[a − t] ˆt [a − t] [a − c2] = some scalar.
We use the identity uv ≡ 2u ∧ u + vu to rewrite that result as
2 [a − t] ∧ ˆt + ˆt [a − t] [a − t] [a − c2] = some scalar.
We expand that result as
2a ∧ ˆt [a − t] [a − c2] − (a − t)
2 ˆt [a − c2] = some scalar,
from which
2a ∧ ˆt [a − t] [a − c2] − (a − t)
2 ˆt [a − c2] 2 = 0.
After further expansions and simplifications, we obtain
a2
− r1
2
− 2a · c2 + 2r1 (r1 + r2) = 0. (3)
Let’s pause now to compare that result to
p2
− r1
2
− 2p · c3 + 2t · c3 = 0,
which was obtained in [6] for the CCP limiting case. The geometric elements
referred to therein are shown in Fig. 3 .
Returning now to the present (CPP) limiting case, we continue by recog-
nizing that c2 = (r1 + r2)ˆt, then substituting in Eq. (3) the expression for
r1 + r2 that’s given in Eq. (2):
a2
− r1
2
− 2a ·
2r1
2
− a2
− b2
4r1 − 2 (a + b) · ˆt
ˆt + 2r1
2r1
2
− a2
− b2
4r1 − 2 (a + b) · ˆt
= 0
3](https://image.slidesharecdn.com/apolloniuscppviar2-160925231827/85/An-additional-brief-solution-of-the-CPP-limiting-case-of-the-Problem-of-Apollonius-via-Geometric-Algebra-GA-3-320.jpg)
![Figure 3: The geometric elements used in [6]’s solution of the CCP limiting
case.
After another round of expansions, simplifications, and rearrangements, we ar-
rive at
b2
− r1
2
a − a2
− r1
2
b · ˆt = r1 b2
− a2
,
both sides of which we multiply by r1, giving the result that was obtained in
several ways in [1]:
b2
− r1
2
a − a2
− r1
2
b · t = r1
2
b2
− a2
.
As we know from [1]-[7], a solution of that form means that there are two
solution circles, whose points of tangency are reflections of each other with
respect to the vector w = b2
− r1
2
a − a2
− r1
2
b (Fig. 4). The projections
upon ˆw of the vectors to those points of tangency are equal, and are given by
P ˆw (t) =
r1
2
b2
− a2
w
ˆw.
References
[1] J. Smith, “Rotations of Vectors Via Geometric Algebra: Explanation, and
Usage in Solving Classic Geometric ‘Construction’ Problems” (Version of
11 February 2016). Available at http://vixra.org/abs/1605.0232 .
[2] “Solution of the Special Case ‘CLP’ of the Problem of Apollo-
nius via Vector Rotations using Geometric Algebra”. Available at
http://vixra.org/abs/1605.0314.
[3] “The Problem of Apollonius as an Opportunity for Teaching Students to
Use Reflections and Rotations to Solve Geometry Problems via Geometric
(Clifford) Algebra”. Available at http://vixra.org/abs/1605.0233.
4](https://image.slidesharecdn.com/apolloniuscppviar2-160925231827/85/An-additional-brief-solution-of-the-CPP-limiting-case-of-the-Problem-of-Apollonius-via-Geometric-Algebra-GA-4-320.jpg)
![Figure 4: The two solution circles. Their points of tangency are reflections of
each other with respect to the vector w = b2
− r1
2
a − a2
− r1
2
b
.
[4] “A Very Brief Introduction to Reflections in 2D Geometric Alge-
bra, and their Use in Solving ‘Construction’ Problems”. Available at
http://vixra.org/abs/1606.0253.
[5] “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.
[6] “Simplified Solutions of the CLP and CCP Limiting Cases of the Problem
of Apollonius via Vector Rotations using Geometric Algebra”. Available
at http://vixra.org/abs/1608.0217.
[7] “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.
5](https://image.slidesharecdn.com/apolloniuscppviar2-160925231827/85/An-additional-brief-solution-of-the-CPP-limiting-case-of-the-Problem-of-Apollonius-via-Geometric-Algebra-GA-5-320.jpg)
An additional brief solution of the CPP limiting case of the Problem of Apollonius via Geometric Algebra (GA)
- 1. An additional brief solution of the CPP limiting case of the Problem of Apollonius via Geometric Algebra (GA) Jim Smith QueLaMateNoTeMate.webs.com email: nitac14b@yahoo.com September 25, 2016 Abstract This document adds to the collection of GA solutions to plane-geometry problems, most of them dealing with tangency, that are presented in [1]- [7]. Reference [1] presented several ways of solving the CPP limiting case of the Problem of Apollonius. Here, we use ideas from [6] to solve that case in yet another way. 1 Statement of the problem The CPP limiting case of the Problem of Apollonius reads, “Given a circle and two points outside of it, construct the circles that are tangent to the given one, and that also pass through both of the given points” (Fig. 1). 2 Expressing elements of the problem in ways that facilitate solution via GA We should note that instead of the angles shown, we could have used angles of rotation from b − t to ˆt, and from b − c2 to ˆt. Experience gained from earlier work (especially [6]) suggests that we use the elements identified in Fig. 2. Note, especially, that the angles θ and 2θ terminate at the line that connects the centers of the two circles, and have as their vertices the point of tangency and the center of the solution circle. 1
- 2. Figure 1: The CPP limiting case of the Problem of Apollonius: Given a circle and two points outside of it, construct the circles that are tangent to the given one, and that also pass through both of the given points. Figure 2: The point of tangency t, and the geometric elements of the problem that will be used to identify t via GA. 3 Solution We will follow [6] in beginning by deriving an expression for r2 and r1+r2, which we will then use in an equation that equates two expressions for the multivector e2θi . Please see [6] for further details. 3.1 Expressions for r2 and r1 + r2 From Fig. 2, we can derive that (r1 + r2)ˆt = a + b 2 + (a − b) i a − b r2 2 − a − b 2 2 . We’ll rearrange that as (r1 + r2)ˆt − a + b 2 = (a − b) i a − b r2 2 − a − b 2 2 , 2
- 3. then square both sides. After simplifying and solving for r2, we find that r2 = 2r1 (a + b) · ˆt − a2 − b2 − 2r1 2 4r1 − 2 (a + b) · ˆt . (1) From that result, we can then derive r1 + r2 = 2r1 2 − a2 − b2 4r1 − 2 (a + b) · ˆt . (2) 3.2 Equating two expressions for the multivector e2θi Note that of the two given points, only one of them (in this case, a) figures in multivector that we are using. We make no use of the other point, than to develop an expression for r2 in Eq. (3.1). Still following [6], we see from Fig. 2 that a − t a − t ˆt =eθi a − t a − t ˆt =eθi = a − c2 a − c2 ˆt =e2θi , from which [a − t] ˆt [a − t] [a − c2] = some scalar. We use the identity uv ≡ 2u ∧ u + vu to rewrite that result as 2 [a − t] ∧ ˆt + ˆt [a − t] [a − t] [a − c2] = some scalar. We expand that result as 2a ∧ ˆt [a − t] [a − c2] − (a − t) 2 ˆt [a − c2] = some scalar, from which 2a ∧ ˆt [a − t] [a − c2] − (a − t) 2 ˆt [a − c2] 2 = 0. After further expansions and simplifications, we obtain a2 − r1 2 − 2a · c2 + 2r1 (r1 + r2) = 0. (3) Let’s pause now to compare that result to p2 − r1 2 − 2p · c3 + 2t · c3 = 0, which was obtained in [6] for the CCP limiting case. The geometric elements referred to therein are shown in Fig. 3 . Returning now to the present (CPP) limiting case, we continue by recog- nizing that c2 = (r1 + r2)ˆt, then substituting in Eq. (3) the expression for r1 + r2 that’s given in Eq. (2): a2 − r1 2 − 2a · 2r1 2 − a2 − b2 4r1 − 2 (a + b) · ˆt ˆt + 2r1 2r1 2 − a2 − b2 4r1 − 2 (a + b) · ˆt = 0 3
- 4. Figure 3: The geometric elements used in [6]’s solution of the CCP limiting case. After another round of expansions, simplifications, and rearrangements, we ar- rive at b2 − r1 2 a − a2 − r1 2 b · ˆt = r1 b2 − a2 , both sides of which we multiply by r1, giving the result that was obtained in several ways in [1]: b2 − r1 2 a − a2 − r1 2 b · t = r1 2 b2 − a2 . As we know from [1]-[7], a solution of that form means that there are two solution circles, whose points of tangency are reflections of each other with respect to the vector w = b2 − r1 2 a − a2 − r1 2 b (Fig. 4). The projections upon ˆw of the vectors to those points of tangency are equal, and are given by P ˆw (t) = r1 2 b2 − a2 w ˆw. References [1] J. Smith, “Rotations of Vectors Via Geometric Algebra: Explanation, and Usage in Solving Classic Geometric ‘Construction’ Problems” (Version of 11 February 2016). Available at http://vixra.org/abs/1605.0232 . [2] “Solution of the Special Case ‘CLP’ of the Problem of Apollo- nius via Vector Rotations using Geometric Algebra”. Available at http://vixra.org/abs/1605.0314. [3] “The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via Geometric (Clifford) Algebra”. Available at http://vixra.org/abs/1605.0233. 4
- 5. Figure 4: The two solution circles. Their points of tangency are reflections of each other with respect to the vector w = b2 − r1 2 a − a2 − r1 2 b . [4] “A Very Brief Introduction to Reflections in 2D Geometric Alge- bra, and their Use in Solving ‘Construction’ Problems”. Available at http://vixra.org/abs/1606.0253. [5] “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. [6] “Simplified Solutions of the CLP and CCP Limiting Cases of the Problem of Apollonius via Vector Rotations using Geometric Algebra”. Available at http://vixra.org/abs/1608.0217. [7] “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. 5