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Mathematics 309Part I - The elementary geometry of wave motion3. Affine functionsTwo dimensionsAn affine function in 2D is a scalar-valued function of the form Ax+By+C. The level curves of the function are the curves where the function takes a fixed value. For example, if the function is x+y then some of the level lines look like this:
Of course the level lines of the function Ax+By are the same as those of Ax+BY+C, but with different constants. So we may as well assume C=0, and the level lines are of the form Ax+By=k. If the coefficient B is not 0 then the equation Ax+By=k can be solved to get the equation y = (k-Ax)/B, the more familiar equation from elementary mathematics. From this equation we can figure out the slope and the y-intercept, and then plot the curve, which is actually a straight line. The level curves that escape thsi are those where B=0, or x=k/A, which are vertical. But it is unnecessary to distinguish cases. Here we want to explain more directly the geometric properties of the level curves.
The line x+y=1 is perpendicular to [1,1], on the same side of ther origin as [1,1], and at distance 1/ from the origin. < Why are these things true?The first on eis easy. If P = (P_{x}, P_{y}) Q = (Q_{x}, Q_{y}) are points on the line AX+By=k, then
A P_{x}+B P_{y}=k
which means that the dot-product of [A,B] and P-Q equals 0, or that they are perpendicular. Since P-Q lies in the direction of the line, that's the first assertion. There will be a unique scalar multiple t[A,B] which lies on the line Ax+By=k. This will be the case when
A(tA)+B(tB) = t(A^{2}+B^{2}) = k
The length of the vector t[A,B] is then k/| [A,B] |, and this proves the second assertion. The last one follows from the sign of t, i.e. of k. Three dimensionsAn affine function is of the form Ax+By+Cz+D. The points (x,y,z) where this has a constant value is a plane Ax+By+Cz+D=k. Here:
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