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I am a high school student writing a maths exploration paper. For this paper I wanted to be able to find the point in space on a curved line, for instance a circle. Is it possible to obtain a vector equation for this or do alternate methods exist to model this?

I have done preliminary research on the internet, yet I was not able to find any other maths that could help me with this.

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    $\begingroup$ So you want the equation of a circle in two dimensions, described by vectors? Well, if you have an origin $O$ and a point $P$ on the circle, then all points $P$ that satisfy $$ \left| \vec{OP} \right| = 1 $$ are on a unit circle. $\endgroup$
    – Matti P.
    Commented Aug 12, 2019 at 11:47
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    $\begingroup$ You might also be interested in tangent planes to differentiable curves. In which case, the tangent plane can be found directly from the partial derivatives of the function at the point. $\endgroup$ Commented Aug 12, 2019 at 11:49
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    $\begingroup$ That sounds like a good idea, but you would have to explain to make it fit a high school level. $\endgroup$
    – Toby Mak
    Commented Aug 12, 2019 at 11:51
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    $\begingroup$ @MattiP. the circle would be in a 3D space as I am creating a model of the international space station orbiting the Earth. $\endgroup$
    – Liam
    Commented Aug 12, 2019 at 11:58
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    $\begingroup$ @Liam You might want to use Kepler's Laws then. It'll allow you to avoid the calculus of elliptical flights. $\endgroup$ Commented Aug 12, 2019 at 11:59

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A typical curve $\gamma$ in ${\mathbb R}^3$ is described by a function $$\gamma:\quad[a,b]\to{\mathbb R}^3,\qquad t\mapsto{\bf r}(t)=\bigl(x(t),y(t),z(t)\bigr)\ .$$ such a thing is called a parametric representation of $\gamma$. If your $\gamma$ is a circle you have to know the center ${\bf c}=(c_1,c_2,c_3)$ and the radius $r>0$ of this circle, and above all: the unit normal ${\bf n}$ of the plane through ${\bf c}$ containing this circle. Given ${\bf n}$ it is a problem of linear algebra to construct two mutually orthogonal unit vectors ${\bf u}=(u_1,u_2,u_3)$ and ${\bf v}=(v_1,v_2,v_3)$ in this plane. When you have found one such vector ${\bf u}\perp{\bf n}$ you can put ${\bf v}:={\bf n}\times{\bf u}$. After ${\bf u}$ and ${\bf v}$ have been constructed a "vectorial" parametric representation of your circle $\gamma$ would be $$\gamma:\quad[0,2\pi]\to{\mathbb R}^3,\qquad t\mapsto {\bf c}+r\bigl(\cos t\> {\bf u}+\sin t\>{\bf v}\bigr)\ .$$ Of course you can write this out in the three coordinates separately.

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