well - think of "lifting" (the starting part) the gradient vector in the xy-plane "up" to the surface:
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Expression 23: left parenthesis, 1 minus "t" , 2 left parenthesis, 1 minus "t" , right parenthesis , "f" left parenthesis, 1 minus "t" , 2 left parenthesis, "t" minus 1 , right parenthesis , right parenthesis , right parenthesis1−t,21−t,f1−t,2t−1
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Lesson learned: ...let's talk thru this ...
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The gradient vector points in the direction the z=f(x,y) is increasing most rapidly.
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Observe (to come next):
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Let's go to 2D view
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The (3D) plane that is
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* thru the point (1,2,0) and
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* with normal vector < f_x (1,2) , f_y (1,2) , 0 > is:
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has the equation
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Expression 32: "f" Subscript, "x" , Baseline left parenthesis, 1 , 2 , right parenthesis left parenthesis, "x" minus 1 , right parenthesis positive "f" Subscript, "y" , Baseline left parenthesis, 1 , 2 , right parenthesis left parenthesis, "y" minus 2 , right parenthesis positive 0 left parenthesis, "z" minus 0 , right parenthesis equals 0 left brace, "z" equals 0 , right bracefx1,2x−1+fy1,2y−2+0z−0=0z=0
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... where the {z=0} is restricting the plane ... can explore w/out if wanted ...
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Expression 34: "f" Subscript, "x" , Baseline left parenthesis, 1 , 2 , right parenthesis left parenthesis, "x" minus 1 , right parenthesis positive "f" Subscript, "y" , Baseline left parenthesis, 1 , 2 , right parenthesis left parenthesis, "y" minus 2 , right parenthesis positive 0 left parenthesis, "z" minus 0 , right parenthesis equals 0fx1,2x−1+fy1,2y−2+0z−0=0
Expression 39: "C" Subscript, "p" "i" "n" "k" , Baseline equals hsv left parenthesis, 310 , 0.8 , 1 , right parenthesisCpink=hsv310,0.8,1
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Expression 40: "f" Subscript, "x" , Baseline left parenthesis, "x" , "y" , right parenthesis equals negative StartFraction, "x" Over StartRoot, 9 minus "x" squared minus "y" squared , EndRoot , EndFractionfxx,y=−x9−x2−y2
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Expression 41: "f" Subscript, "y" , Baseline left parenthesis, "x" , "y" , right parenthesis equals negative StartFraction, "y" Over StartRoot, 9 minus "x" squared minus "y" squared , EndRoot , EndFractionfyx,y=−y9−x2−y2
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Expression 42: "f" Subscript, "x" , Baseline left parenthesis, 1 , 2 , right parenthesisfx1,2
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Expression 43: "f" Subscript, "y" , Baseline left parenthesis, 1 , 2 , right parenthesisfy1,2
