Quote:
Originally Posted by Robh
I didn't find any reference to this constancy, which is patently ridiculous as sqr(x1^2+y1^2)+sqr(x2^2+y2^2)<>sqr( (x1+x2)^2+(y1+y2)^2). The fact is that the length of any curve can be obtained by integrating ds where ds^2=dx^2+dy^2.
dx and dy can be considered the horizontal and vertical components of an infinitesimally small right triangle drawn on the section of curve. His assumption that ds=dx+dy is a clear contradiction of Pythagoras' Theorem. As I mentioned before ds<dx+dy for all right-triangles on the curve. His assumptions are unjustifiable.
Regards, Rob
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The flaw with Mathis is that he is calculating the length of the
chord AC, not the
arc AC.
This is clear from his geometrical construction.
For example the chord AC= (AD^2+DC^2)^0.5 is for the triangle ADC.
The chord AC is an approximation of the arc length using a single triangle.
For simplicity lets subdivide into equal subtriangles.
A subtriangle can be constructed with sides AD/n, DC/n where n is the number of subtriangles used. In this case 1/n is a scale factor.
The hypotenuse for a subtriangle is y= ((AD/n)^2 + (DC/n)^2)^0.5.
The total length is the sum of the hypotenuses = n*[((AD/n)^2 + (DC/n)^2)^0.5].
But n*[((AD/n)^2 + (DC/n)^2)^0.5] = (AD^2+DC^2)^0.5 which is the length of the chord AC not the length of the arc AC.
Hence for any value n the result is the same.
Mathis doesn't know his chords from his arcs.
Regards
Steven