What is the angle in degrees? (Try 2)
The problem's question: What is the angle in degrees?
My errors in the previous version were that I did some copy-and-pastes wrong when assembling the problem text and I used random selection with replacement when I intended random selection without replacement when selecting points on the surface of the 4D sphere. If it helps, then, the radius of the sphere is the same as in the previous version.
The points on the surface of the sphere: {25, 113, 110, 107}, {112, 185, 119, 107}, {121, 104, 103, 187}, {193, 98, 111, 91}, and {193, 104, 119, 107}.
Two points on the infinite line: {277, 109, 83, 109}, {361, 113, 71, 112}.
The infinite line intersects the sphere at two points. With {121, 104, 103, 187} being the vertex of the angle and two lines passing through the vertex and each point of intersection respectively, what is the (smaller, if necessary) angle between the lines at the vertex?
The answer is 90.
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1 solution
There are many ways to solve this problem:
The first part of the problem is to get the center and radius of the sphere . This is the hardest part.
The next part is to discover where the infinite line meets interesting parts of the sphere. I find it interesting that the infinite line passes through the center of the sphere. The infinite line is a diameter of of the sphere!
The last part is finding the angle. The vertex of the angle is one of the points given as being on the surface of the sphere. By Thales' Theorem , the angle is a right angle or 90 degrees.
Now, the implementation:
Circumsphere [ { { 2 5 , 1 1 3 , 1 1 0 , 1 0 7 } , { 1 1 2 , 1 8 5 , 1 1 9 , 1 0 7 } , { 1 2 1 , 1 0 4 , 1 0 3 , 1 8 7 } , { 1 9 3 , 9 8 , 1 1 1 , 9 1 } , { 1 9 3 , 1 0 4 , 1 1 9 , 1 0 7 } } ] ⇒ Sphere [ { 1 0 9 , 1 0 1 , 1 0 7 , 1 0 3 } , 8 5 ]
Points along the infinite line: { − 3 , { − 1 4 3 , 8 9 , 1 4 3 , 9 4 } } { − 2 , { − 5 9 , 9 3 , 1 3 1 , 9 7 } } { − 1 , { 2 5 , 9 7 , 1 1 9 , 1 0 0 } } { 0 , { 1 0 9 , 1 0 1 , 1 0 7 , 1 0 3 } } { 1 , { 1 9 3 , 1 0 5 , 9 5 , 1 0 6 } } { 2 , { 2 7 7 , 1 0 9 , 8 3 , 1 0 9 } } { 3 , { 3 6 1 , 1 1 3 , 7 1 , 1 1 2 } }
The points labeled -1 and 1 are the intersection points. The point labeled 0 is the center of the sphere.
If any of the surface points on the sphere, including those listed in the problem specification, excluding the two points on the infinite line are substituted for p in the formula below, then, the value of the expression will be 0 , which show a right angle everywhere except as previously noted:
( { 9 7 , 1 9 , 1 1 9 , − 1 1 2 } − p ) . ( { 1 0 5 , 1 8 7 , 9 5 , − 1 0 6 } − p ) ⇒ 0
which also shows that the angle is a right angle.
To get most values to be integers, the points on the infinite line were 2 and 3 radii from the center, a fact which could be have been determined by computing the Euclidean distance between the center and the specified points. Once that is noticed, computing the intersection points is trivial as demonstrated above.