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Thread: Linear Interpolation Question

  1. #1
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    Linear Interpolation Question

    Hello:
    The formula: y1 * (1-x) + y2 * x is a Linear Interpolation formula. I am trying to understand if this formula will yield the middle values between two points such as: Point 1: (5,10 ) and Point 2: (20,15)

    Would this be the diagonal line connecting these two points. If so, is "x" the range value from x1 to x2?. I am trying to code this in a computer function to test.

    Thanks in advance

    Mark
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  2. #2
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    Linear interpolation

    Hello mhartman

    Welcome to Math Help Forum!
    Quote Originally Posted by mhartman View Post
    Hello:
    The formula: y1 * (1-x) + y2 * x is a Linear Interpolation formula. I am trying to understand if this formula will yield the middle values between two points such as: Point 1: (5,10 ) and Point 2: (20,15)

    Would this be the diagonal line connecting these two points. If so, is "x" the range value from x1 to x2?. I am trying to code this in a computer function to test.

    Thanks in advance

    Mark
    I'm not sure where the formula you quote comes from, but this may help.

    The midpoint of the line joining the points $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2, y_2)$ is

    $\displaystyle \Big(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\Big)$

    So in the example you quote, it will be $\displaystyle \Big(\frac{5+20}{2}, \frac{10+15}{2}\Big) = (12.5, 12.5)$.

    The equation of the line joining these points is:

    $\displaystyle \frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$

    which can be re-arranged as:

    $\displaystyle y = y_1 + (x-x_1)\frac{y_2-y_1}{x_2-x_1}$

    which gives you the linear interpolation formula that will calculate the value of $\displaystyle y$ for any value of $\displaystyle x$ in the interval $\displaystyle (x_1, x_2)$. So, for example, in the interval you quote, if you want the value of y when $\displaystyle x = 13$, it's

    $\displaystyle y = 10 + (13-5)\frac{15-10}{20-5} = 10 + 8\times\frac{1}{3}= 12\tfrac23$

    Grandad
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  3. #3
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    Thank you so much for taking the time to reply.

    Mark
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