Maximum area of a triangle formed from the equidistant lines of 4 different points
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
The attached diagram shows a Triangle $abc$ formed from the connection of points $M, I_1, I_2, I_3$ such that some $2D$ space is shared equally between the points.
I would like to find the maximum area of the triangle defined by the lines $a, b, c$.
From Heron's formula the Area, $mathcalA$ of the triangle with lines $a,b$ and $c$ is
$$mathcalA = sqrts(s-a)(s-b)(s-c),, textfor: s = fraca+b+c2$$
And on further expansion:
$$mathcalA = fracsqrt2a^2(b^2+c^2)-(b^2-c^2)^2-a^44$$
From the diagram shown, the sides of the triangle $a,b,c$ are obtained from $d_1, d_2, d_3, omega_1, omega_2, omega_3$ thus:
$$a = frac12(sqrtfracd_1^2-2d_1d_2cos(omega_1-omega_2)+d_2^21-cos^2(omega_1-omega_2))\
b = frac12(sqrtfracd_2^2-2d_2d_3cos(omega_2-omega_3)+d_3^21-cos^2(omega_2-omega_3))\c = sqrt[frac[d_1sinomega_2-d_2sinomega_1]sin(omega_3-omega_2)-[d_2sinomega_3-d_3sinomega_2]sin(omega_2-omega_1)2sin(omega_2-omega_1)sin(omega_3-omega_2)]^2 + [frac[d_1cosomega_2-d_2cosomega_1]sin(omega_2-omega_3)-[d_2cosomega_3-d_3cosomega_2]sin(omega_1-omega_2)2sin(omega_1-omega_2)sin(omega_2-omega_3)]^2$$
The problems:
Task 1: Maximze $mathcalA(omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
Task 2: Maximze $mathcalA(omega_2, omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
I do not have much experience in optimisation, hence unsure of how to go about the problems
geometry optimization
edited Aug 26 at 14:39
amWhy
190k26221433
asked Aug 26 at 12:19
A. Ethan
62
add a comment |Â
up vote
1
down vote
favorite
The attached diagram shows a Triangle $abc$ formed from the connection of points $M, I_1, I_2, I_3$ such that some $2D$ space is shared equally between the points.
I would like to find the maximum area of the triangle defined by the lines $a, b, c$.
From Heron's formula the Area, $mathcalA$ of the triangle with lines $a,b$ and $c$ is
$$mathcalA = sqrts(s-a)(s-b)(s-c),, textfor: s = fraca+b+c2$$
And on further expansion:
$$mathcalA = fracsqrt2a^2(b^2+c^2)-(b^2-c^2)^2-a^44$$
From the diagram shown, the sides of the triangle $a,b,c$ are obtained from $d_1, d_2, d_3, omega_1, omega_2, omega_3$ thus:
$$a = frac12(sqrtfracd_1^2-2d_1d_2cos(omega_1-omega_2)+d_2^21-cos^2(omega_1-omega_2))\
b = frac12(sqrtfracd_2^2-2d_2d_3cos(omega_2-omega_3)+d_3^21-cos^2(omega_2-omega_3))\c = sqrt[frac[d_1sinomega_2-d_2sinomega_1]sin(omega_3-omega_2)-[d_2sinomega_3-d_3sinomega_2]sin(omega_2-omega_1)2sin(omega_2-omega_1)sin(omega_3-omega_2)]^2 + [frac[d_1cosomega_2-d_2cosomega_1]sin(omega_2-omega_3)-[d_2cosomega_3-d_3cosomega_2]sin(omega_1-omega_2)2sin(omega_1-omega_2)sin(omega_2-omega_3)]^2$$
The problems:
Task 1: Maximze $mathcalA(omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
Task 2: Maximze $mathcalA(omega_2, omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
I do not have much experience in optimisation, hence unsure of how to go about the problems
geometry optimization
edited Aug 26 at 14:39
amWhy
190k26221433
asked Aug 26 at 12:19
A. Ethan
62
What does $mathcalA(omega_1)$ and $mathcalA(omega_2,omega_3)$ mean?
– John Glenn
Aug 26 at 14:39
@JohnGlenn Thank you. $mathcalA(omega_3)$ means $mathcalA$ is a function of $omega_3$ and the goal is to find the value of $omega_3$ that maximises $mathcalA$. Similarly, $mathcalA(omega_2,omega_3)$ implies the optimisation problem is to find the values of $omega_2$ and $omega_3$ that maximise $mathcalA$.
– A. Ethan
Aug 26 at 15:09
What is the difference with the Voronoi distance between the points?
– Cesareo
Aug 27 at 20:07
@Cesareo Thank you. True, the formulae I presented help to generate the Voronoi distances between the given points. However, the optimisation problems are to help me investigate the effect of changing the positions of $I_2, I_3$, using $omega_2, omega_3$ with the assumption that $d_1, d_2, d_3, omega_1$ are fixed. The task is not to generate the Voronoi distances but it is an optimisation problem
– A. Ethan
Aug 28 at 8:30
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The attached diagram shows a Triangle $abc$ formed from the connection of points $M, I_1, I_2, I_3$ such that some $2D$ space is shared equally between the points.
I would like to find the maximum area of the triangle defined by the lines $a, b, c$.
From Heron's formula the Area, $mathcalA$ of the triangle with lines $a,b$ and $c$ is
$$mathcalA = sqrts(s-a)(s-b)(s-c),, textfor: s = fraca+b+c2$$
And on further expansion:
$$mathcalA = fracsqrt2a^2(b^2+c^2)-(b^2-c^2)^2-a^44$$
From the diagram shown, the sides of the triangle $a,b,c$ are obtained from $d_1, d_2, d_3, omega_1, omega_2, omega_3$ thus:
$$a = frac12(sqrtfracd_1^2-2d_1d_2cos(omega_1-omega_2)+d_2^21-cos^2(omega_1-omega_2))\
b = frac12(sqrtfracd_2^2-2d_2d_3cos(omega_2-omega_3)+d_3^21-cos^2(omega_2-omega_3))\c = sqrt[frac[d_1sinomega_2-d_2sinomega_1]sin(omega_3-omega_2)-[d_2sinomega_3-d_3sinomega_2]sin(omega_2-omega_1)2sin(omega_2-omega_1)sin(omega_3-omega_2)]^2 + [frac[d_1cosomega_2-d_2cosomega_1]sin(omega_2-omega_3)-[d_2cosomega_3-d_3cosomega_2]sin(omega_1-omega_2)2sin(omega_1-omega_2)sin(omega_2-omega_3)]^2$$
The problems:
Task 1: Maximze $mathcalA(omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
Task 2: Maximze $mathcalA(omega_2, omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
I do not have much experience in optimisation, hence unsure of how to go about the problems
geometry optimization
edited Aug 26 at 14:39
amWhy
190k26221433
asked Aug 26 at 12:19
A. Ethan
62
The attached diagram shows a Triangle $abc$ formed from the connection of points $M, I_1, I_2, I_3$ such that some $2D$ space is shared equally between the points.
I would like to find the maximum area of the triangle defined by the lines $a, b, c$.
From Heron's formula the Area, $mathcalA$ of the triangle with lines $a,b$ and $c$ is
$$mathcalA = sqrts(s-a)(s-b)(s-c),, textfor: s = fraca+b+c2$$
And on further expansion:
$$mathcalA = fracsqrt2a^2(b^2+c^2)-(b^2-c^2)^2-a^44$$
From the diagram shown, the sides of the triangle $a,b,c$ are obtained from $d_1, d_2, d_3, omega_1, omega_2, omega_3$ thus:
$$a = frac12(sqrtfracd_1^2-2d_1d_2cos(omega_1-omega_2)+d_2^21-cos^2(omega_1-omega_2))\
b = frac12(sqrtfracd_2^2-2d_2d_3cos(omega_2-omega_3)+d_3^21-cos^2(omega_2-omega_3))\c = sqrt[frac[d_1sinomega_2-d_2sinomega_1]sin(omega_3-omega_2)-[d_2sinomega_3-d_3sinomega_2]sin(omega_2-omega_1)2sin(omega_2-omega_1)sin(omega_3-omega_2)]^2 + [frac[d_1cosomega_2-d_2cosomega_1]sin(omega_2-omega_3)-[d_2cosomega_3-d_3cosomega_2]sin(omega_1-omega_2)2sin(omega_1-omega_2)sin(omega_2-omega_3)]^2$$
The problems:
Task 1: Maximze $mathcalA(omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
Task 2: Maximze $mathcalA(omega_2, omega_3)$ subject to the following constraints
a. $-180leq omega_1,omega_2,omega_3 leq 180$
b. $omega_3 > omega_2 > omega_1$
c. $0.4 leq d1,d2,d3 leq 1.6$
d. $0.4 leq sqrtd_1^2+d_2^2-2d_1 d_2cos(w_1-w_2), sqrtd_2^2+d_3^2-2d_2 d_3cos(w_2-w_3),\ sqrtd_1^2+d_3^2-2d_1 d_3cos(w_1-w_3), leq 1.6$
I do not have much experience in optimisation, hence unsure of how to go about the problems
geometry optimization
edited Aug 26 at 14:39
amWhy
190k26221433
asked Aug 26 at 12:19
A. Ethan
62
edited Aug 26 at 14:39
amWhy
190k26221433
edited Aug 26 at 14:39
amWhy
190k26221433
edited Aug 26 at 14:39
amWhy
190k26221433
190k26221433
asked Aug 26 at 12:19
A. Ethan
62
asked Aug 26 at 12:19
A. Ethan
62
asked Aug 26 at 12:19
A. Ethan
62
62
What does $mathcalA(omega_1)$ and $mathcalA(omega_2,omega_3)$ mean?
– John Glenn
Aug 26 at 14:39
@JohnGlenn Thank you. $mathcalA(omega_3)$ means $mathcalA$ is a function of $omega_3$ and the goal is to find the value of $omega_3$ that maximises $mathcalA$. Similarly, $mathcalA(omega_2,omega_3)$ implies the optimisation problem is to find the values of $omega_2$ and $omega_3$ that maximise $mathcalA$.
– A. Ethan
Aug 26 at 15:09
What is the difference with the Voronoi distance between the points?
– Cesareo
Aug 27 at 20:07
@Cesareo Thank you. True, the formulae I presented help to generate the Voronoi distances between the given points. However, the optimisation problems are to help me investigate the effect of changing the positions of $I_2, I_3$, using $omega_2, omega_3$ with the assumption that $d_1, d_2, d_3, omega_1$ are fixed. The task is not to generate the Voronoi distances but it is an optimisation problem
– A. Ethan
Aug 28 at 8:30
add a comment |Â
What does $mathcalA(omega_1)$ and $mathcalA(omega_2,omega_3)$ mean?
– John Glenn
Aug 26 at 14:39
@JohnGlenn Thank you. $mathcalA(omega_3)$ means $mathcalA$ is a function of $omega_3$ and the goal is to find the value of $omega_3$ that maximises $mathcalA$. Similarly, $mathcalA(omega_2,omega_3)$ implies the optimisation problem is to find the values of $omega_2$ and $omega_3$ that maximise $mathcalA$.
– A. Ethan
Aug 26 at 15:09
What is the difference with the Voronoi distance between the points?
– Cesareo
Aug 27 at 20:07
@Cesareo Thank you. True, the formulae I presented help to generate the Voronoi distances between the given points. However, the optimisation problems are to help me investigate the effect of changing the positions of $I_2, I_3$, using $omega_2, omega_3$ with the assumption that $d_1, d_2, d_3, omega_1$ are fixed. The task is not to generate the Voronoi distances but it is an optimisation problem
– A. Ethan
Aug 28 at 8:30
What does $mathcalA(omega_1)$ and $mathcalA(omega_2,omega_3)$ mean?
– John Glenn
Aug 26 at 14:39
What does $mathcalA(omega_1)$ and $mathcalA(omega_2,omega_3)$ mean?
– John Glenn
Aug 26 at 14:39
@JohnGlenn Thank you. $mathcalA(omega_3)$ means $mathcalA$ is a function of $omega_3$ and the goal is to find the value of $omega_3$ that maximises $mathcalA$. Similarly, $mathcalA(omega_2,omega_3)$ implies the optimisation problem is to find the values of $omega_2$ and $omega_3$ that maximise $mathcalA$.
– A. Ethan
Aug 26 at 15:09
@JohnGlenn Thank you. $mathcalA(omega_3)$ means $mathcalA$ is a function of $omega_3$ and the goal is to find the value of $omega_3$ that maximises $mathcalA$. Similarly, $mathcalA(omega_2,omega_3)$ implies the optimisation problem is to find the values of $omega_2$ and $omega_3$ that maximise $mathcalA$.
– A. Ethan
Aug 26 at 15:09
What is the difference with the Voronoi distance between the points?
– Cesareo
Aug 27 at 20:07
What is the difference with the Voronoi distance between the points?
– Cesareo
Aug 27 at 20:07
@Cesareo Thank you. True, the formulae I presented help to generate the Voronoi distances between the given points. However, the optimisation problems are to help me investigate the effect of changing the positions of $I_2, I_3$, using $omega_2, omega_3$ with the assumption that $d_1, d_2, d_3, omega_1$ are fixed. The task is not to generate the Voronoi distances but it is an optimisation problem
– A. Ethan
Aug 28 at 8:30
@Cesareo Thank you. True, the formulae I presented help to generate the Voronoi distances between the given points. However, the optimisation problems are to help me investigate the effect of changing the positions of $I_2, I_3$, using $omega_2, omega_3$ with the assumption that $d_1, d_2, d_3, omega_1$ are fixed. The task is not to generate the Voronoi distances but it is an optimisation problem
– A. Ethan
Aug 28 at 8:30
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2894981%2fmaximum-area-of-a-triangle-formed-from-the-equidistant-lines-of-4-different-poin%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
What does $mathcalA(omega_1)$ and $mathcalA(omega_2,omega_3)$ mean?
– John Glenn
Aug 26 at 14:39
@JohnGlenn Thank you. $mathcalA(omega_3)$ means $mathcalA$ is a function of $omega_3$ and the goal is to find the value of $omega_3$ that maximises $mathcalA$. Similarly, $mathcalA(omega_2,omega_3)$ implies the optimisation problem is to find the values of $omega_2$ and $omega_3$ that maximise $mathcalA$.
– A. Ethan
Aug 26 at 15:09
What is the difference with the Voronoi distance between the points?
– Cesareo
Aug 27 at 20:07
@Cesareo Thank you. True, the formulae I presented help to generate the Voronoi distances between the given points. However, the optimisation problems are to help me investigate the effect of changing the positions of $I_2, I_3$, using $omega_2, omega_3$ with the assumption that $d_1, d_2, d_3, omega_1$ are fixed. The task is not to generate the Voronoi distances but it is an optimisation problem
– A. Ethan
Aug 28 at 8:30