CODEWITHSUNDEEP
mathvectorgeometrytrigonometry
user1782677
user1782677

Reputation: 2007

Code to find an angle theta

So I have 3 3D Vectors, W, T1 and T2 that satisfy the relationship W = T1*cos(theta) + T2*sin(theta).

I need to come up with an algorithm that can find theta given these 3 vectors. However I'm stuck and don't know where to start even.

Upvotes: 0

Views: 266

Answers (2)

maxim1000
maxim1000

Reputation: 6365

If T1 and T2 are not collinear, you can use cross product:

  • W = T1*cos(theta) + T2*sin(theta)
  • [W,T1]=[T2,T1]*sin(theta)
  • [W,T2]=[T1,T2]*cos(theta)

If they are collinear, just project them on a line and solve scalar equation A=B*cos(theta)+C*sin(theta)

Upvotes: 1

Matthew T. Staebler
Matthew T. Staebler

Reputation: 4996

Use techniques from linear algebra to solve for the possibilities of cos(theta) and sin(theta).

[ T1_1  |  T2_1  |  W_1 ]
[ T1_2  |  T2_2  |  W_2 ]

[ 1     |  T2_1 / T1_1  |  W_1 / T1_1 ]
[ T1_2  |  T2_2         |  W2         ]

[ 1     |  T2_1 / T1_1                |  W_1 / T1_1             ]
[ 0     |  T2_2 - T1_2 * T2_1 / T1_1  |  W2 - T1_2 * W_1 / T1_1 ]

[ 1     |  T2_1 / T1_1                |  W_1 / T1_1                                             ]
[ 0     |  1                          |  (W2 - T1_2 * W_1 / T1_1) / (T2_2 - T1_2 * T2_1 / T1_1) ]

[ 1     |  0                          |  W_1 / T1_1 - T2_1 / T1_1 * (W2 - T1_2 * W_1 / T1_1) / (T2_2 - T1_2 * T2_1 / T1_1)              ]
[ 0     |  1                          |  (W2 - T1_2 * W_1 / T1_1) / (T2_2 - T1_2 * T2_1 / T1_1)                                         ]

So,

cos(theta) = alpha * W_1 / T1_1 - T2_1 / T1_1 * (W2 - T1_2 * W_1 / T1_1) / (T2_2 - T1_2 * T2_1 / T1_1)
sin(theta) = alpha * (W2 - T1_2 * W_1 / T1_1) / (T2_2 - T1_2 * T2_1 / T1_1)

We know that

cos(theta)^2 + sin(theta)^2 = 1

Plugging the previous equations for cos(theta) and sin(theta) into that last equation, we can solve for alpha. Knowing that, we can calculate the actual value of theta by using either arccosine or arcsine.

Note, that I have not checked my work in any of these steps, so I do not make any guarantees about the accuracy of the equations. I leave that as an exercise for you.

Upvotes: 0

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