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Hi,

I've been trying to deduce the general equation of a circle in 3D space, but without much luck.

Well, I'll jump in with what I've got so far.

[tex]P = (X_{P},Y_{P},Z_{P}) = \mbox{General point on diameter of circle circle.}[/tex]

[tex]C = (X_{C},Y_{C},Z_{C}) = \mbox{Centre of the circle.} [/tex]

[tex]\bar{d} = \left(\begin{array}{cc}x_{d}\\y_{d}\\z_{d}\end{array}\right) = \mbox{Perpendicular to the plane of the circle.} [/tex]

[tex]r = \mbox{radius of circle} [/tex]

These are the identities I've used to describe it:

[tex]\vec{PC}[/tex] is perperpendicular to [tex]\bar{d}[/tex] so

[tex]\vec{PC}.\bar{d}=0[/tex]

and,

[tex]|\vec{PC}| = r[/tex]

these expand (somewhat clumsily) to:

[tex]x_{d}(X_{C}-X_{P})+y_{d}(Y_{C}-Y_{P})+z_{d}(Z_{C}-Z_{P})=0 [/tex]

and

[tex]\sqrt{(X_{C}-X_{P})^2+(Y_{C}-Y_{P})^2+(Z_{C}-Z_{P})^2}=r[/tex]

Not sure what to do from here, or if I'm even barking up the right tree, any help would be much appreciated

I've been trying to deduce the general equation of a circle in 3D space, but without much luck.

Well, I'll jump in with what I've got so far.

[tex]P = (X_{P},Y_{P},Z_{P}) = \mbox{General point on diameter of circle circle.}[/tex]

[tex]C = (X_{C},Y_{C},Z_{C}) = \mbox{Centre of the circle.} [/tex]

[tex]\bar{d} = \left(\begin{array}{cc}x_{d}\\y_{d}\\z_{d}\end{array}\right) = \mbox{Perpendicular to the plane of the circle.} [/tex]

[tex]r = \mbox{radius of circle} [/tex]

These are the identities I've used to describe it:

[tex]\vec{PC}[/tex] is perperpendicular to [tex]\bar{d}[/tex] so

[tex]\vec{PC}.\bar{d}=0[/tex]

and,

[tex]|\vec{PC}| = r[/tex]

these expand (somewhat clumsily) to:

[tex]x_{d}(X_{C}-X_{P})+y_{d}(Y_{C}-Y_{P})+z_{d}(Z_{C}-Z_{P})=0 [/tex]

and

[tex]\sqrt{(X_{C}-X_{P})^2+(Y_{C}-Y_{P})^2+(Z_{C}-Z_{P})^2}=r[/tex]

Not sure what to do from here, or if I'm even barking up the right tree, any help would be much appreciated

Last edited: