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Math Help - Formula of a Three-Dimensional Ray

  1. #1
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    Post Formula of a Three-Dimensional Ray

    Hi, this is actually an extracurricular topic I'm interested in, so it just barely fits into the Geometry section, sorry.

    Anyway,
    Apparently there is the formula:
    R(t) = R0 + t * Rd , t > 0

    Of this, I understand that R0 = [0X, 0Y, 0Z] represents the *O*rigin of the ray, as a point.

    Also, Rd = [dX, dY, dZ], meaning the *D*irection of the ray, represented as the direction from the origin to [dX, dY, dZ].

    However, I'm stuck at T. What is T, and can this formula be used to solve for, say, z given x and y?

    My bountiful gratuity!
    YottaFlop
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  2. #2
    Junior Member slider142's Avatar
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    t is the length of the ray as measured along the ray in units of the vector Rd. Ie., t=0 is the origin of the ray. t=1 is a point on the ray 1 Rd unit away from the origin (as measured along the ray), that is it is the length \sqrt{dX^2 + dY^2 + dZ^2} away from R0. t = \pi is the point \pi\sqrt{dX^2 + dY^2 + dZ^2} units away from R0 along the ray, and so forth.
    To see this, look at each point separately. t = 1 means add Rd to R0. t=x means add a vector in the direction of Rd whose length is x times the length of Rd to R0. This is just generating all vectors of all positive lengths in the direction of Rd from R0.
    Physically, you might say that the length of Rd is the "speed" with which a particle is travelling along the ray (for motivation, look at the derivative of this function with respect to t, where you interpret t as time). Mathematically, you may only be interested in Rd's that are unit vectors, so that the point corresponding to t = C is C units away from the origin R0.
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  3. #3
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    RE

    Thank you very much!
    Though, I'm not sure where the radical and exponent come into play.
    Last edited by yottaflop; June 26th 2010 at 10:40 AM.
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  4. #4
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    Quote Originally Posted by yottaflop View Post
    Thank you very much!
    Though, I'm not sure where the radical and exponent come into play.
    The formula used by slider142 yields the length (or the absolute value) of a vector:

    If you have the vector \vec v = (a,b,c) then it's length is calculated by:

    |\vec v|=\sqrt{a^2+b^2+c^2}
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  5. #5
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    RE

    Oh, okay. Thanks.
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