Parametrization of a circle

Question

How I can find a parametrization of this curve $\begin {cases}x^2+y^2+z^2=1 \\ x+y+z=1\end {cases}$ ?

Answer 1

One way to get a parameterization for your two equations in three variables is to let one of the variables be the parameter and solve for the other two. Let's choose $x$ as the parameter. Solving for $y$ in your second equation we get

$$z=1-x-y$$

Substituting that into your first equation we get a quadratic equation in $z$:

$$y^2+(x-1)y+(x^2-x)=0$$

Using the quadratic formula,

$$y=\frac 12\left(1-x\pm\sqrt{1+2x-3x^2}\right)$$

Setting $x=t$ and filling in the other expressions, we get

$$\begin{align} x &= t \\[2ex] y &= \frac 12\left(1-t\pm\sqrt{1+2t-3t^2}\right) \\[2ex] z &= \frac 12\left(1-t\mp\sqrt{1+2t-3t^2}\right) \\[2ex] 0 &\le t\le 1 \end{align}$$

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Another way is to look at the geometry. Use this way if you want a "smooth" and single-valued parameterization.

Your first equation is a sphere of radius $1$ centered at the origin, and the second is a plane. (Both equations are in standard form for their respective types.) We easily find multiple points that are in the intersection, namely $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$, so the intersection is a circle.

The center of the circle is the centroid of those three points, namely $(\frac 13,\frac 13,\frac 13)$. The radius is the distance from that center to any of the points on the circle, namely $\sqrt{\frac 23}$.

So one way to get a parameterization of your curve is to find a parameterization of the circle centered at the origin with radius $\sqrt{\frac 23}$: that is easy. Then find the three points on that circle at $120°$ angles, corresponding to the three points we know on the desired circle. Then find one of the two affine transformations that take the one circle to the target circle. Apply that transformation to the parameterization of the circle, and you are done.

You could also find a circle on the sphere of radius $\sqrt{\frac 23}$ then find a rotation transformation around the origin that takes that circle to your desired circle. Finding that transformation would be somewhat easier than the other way.

(I leave only hints for the last part since people are voting to close this question for lack of work on your part. If you add more of your work, I may finish my answer.)