... polar1
Note that Equations 8 and 11 require knowlege of which quadrant the angle lies in, because ranges over .
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... cone2
double cone means that it is two standard'' cones joined at their apices.
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... equation3
The equation itself is rather fierce and you would not be expected to do the full expansion in an exam. For those who are interested, it looks like this:
t4 ( xD4 + yD4 + zD4 + 2 xD2 yD2 + 2 xD2 zD2 + 2 yD2 zD2 ) + t3 ( 4 xD3 xE + 4 yD3 yE + 4 zD3 zE + 4 xD2 yD yE + 4 xD2 zD zE + 4 xD xE yD2 + 4 yD2 zD zE + 4 xD xE zD2 + 4 yD yE zD2 ) + t2 ( - 2 R2 xD2 - 2 R2 yD2 + 2 R2 zD2 - 2 r2 xD2 - 2 r2 yD2 - 2 r2 zD2 + 6 xD2 xE2 + 2 xE2 yD2 + 8 xD xE yD yE + 2 xD2 yE2 + 6 yD2 yE2 + 2 xE2 zD2 + 2 yE2 zD2 + 8 xD xE zD zE + 8 yD yE zD zE + 2 xD2 zE2 + 2 yD2 zE2 + 6 zD2 zE2 ) + t ( - 4 R2 xD xE - 4 R2 yD yE + 4 R2 zD zE - 4 r2 xD xE - 4 r2 yD yE - 4 r2 zD zE + 4 xD xE3 + 4 xE2 yD yE + 4 xD xE yE2 + 4 yD yE3 + 4 xE2 zD zE + 4 yE2 zD zE + 4 xD xE zE2 + 4 yD yE zE2 + 4 zD zE3 ) + ( R4 - 2 R2 xE2 - 2 R2 yE2 + 2 R2 zE2 + r4 - 2 r2 R2 - 2 r2 xE2 - 2 r2 yE2 - 2 r2 zE2 + xE4 + yE4 + zE4 + 2 xE2 yE2 + 2 xE2 zE2 + 2 yE2 zE2 ) = 0
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... defined4
The disc is also arbitrarily defined if you want a circular disc, but it could be transformed to provide an ellipse.
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... definition5
The sphere has unit radius; the cylinder has unit radius and is aligned with the z-axis; the cone is similarly aligned and has unit slope; the torus is aligned with the z-axis.
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... transformation6
Where is a translation, a rotation, and a scaling.
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... where7
If you are wondering where r has disappeared to, remember that, in a unit sphere, r=1.
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... being8
If you try this at home you will find that you will need to divide through by the factor .
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