Question

find the right circular cylinder whose guiding curve is the Circle through three points (a,0,0), (0,b,0) and (0,0,c); find also the axis of the cylinder
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Answer 1

A²d+c-x=⁶⅞ñöòëàßsúì⁵raised²³

Answer 2

To Find:

• find the right circular cylinder whose curve is the circle through three points (a. 0, 0), (0, b, 0) and (0, 0, c).
• find the axis of the cylinder?

A point in the plane has the coordinates,

              u(a,0,0)+v(0,b,0)+w(0,0,c)=(ua,vb,wc) with u+v+w=1.

Let us express that the center is in the plane and is equidistant from the given points:

           $(ua-a)^{2} + (vb)^{2} + (wc)^{2} = (ua)^{2} + (vb-b)^{2} + (wc)^{2} = (ua)^{2} + (vb)^{2} + (wc-c)^{2}$

or after expansion and simplification,

                           $a^{2} (1-2u) = b^{2} (1-2v) = c^{2} (1-2w)$

 Let this common value be d2.

Then,

                           $d^{2} (\frac{1}{a^{2} } +\frac{1}{b^{2} } +\frac{1}{c^{2} } ) = 3-2 = 1$

and we obtain the coordinates of the center:

                        $ua = \frac{a}{2} - \frac{d^{2} }{2a} \\vb = \frac{b}{2} - \frac{d^{2} }{2b} \\wc = \frac{c}{2} - \frac{d^{2} }{2c}$,

                           

And the direction of the axis is $(\frac{1}{a},\frac{1}{b},\frac{1}{b})$. The unit vector in the direction of the axis is t =d$(\frac{1}{a},\frac{1}{b},\frac{1}{b})$.

The implicit equation of the cylinder expresses that the distance of a point to the axis is the radius, i.e.

                            ║(x−ua,y−vb,z−wc)×t ║=r.