Demos with Desmos

This famous method maps points on the unit circle onto the real number line. For a given point on the circle (denoted x), we draw a ray from the top of the circle (the ∞ point) through x. We denoted the intersection of this ray with the horizontal axis by S(x), and the mapping x --> S(x) is the stereographic projection.

Notice that the ∞  point at the top of the circle has no associated number S(∞) on the number line, since we can't draw a ray with just one point. However, we can define a new number just for this point, and call it ∞. The real number line with the ∞ point added is called the one-point compactification of the real number line. 

This trick works in higher dimensions as well. You can map a 2D sphere (minus the north pole) onto the 2D plane. More generally an (n-1)-sphere (minus a point) can be mapped onto the (n-1)-dimensional hyperplane!

A cubic is a special name for a degree 3 polynomial. In terms of one variable, a cubic is a sum of the terms 1, x, x², x³ with some coefficients attached.  We often draw curves in the plane by writing y=f(x), for some function f, and then the curve is the set of all pairs (x,y) such that the equation y=f(x) holds. We generalize this by moving y to the right hand side and redefining the curve to be the set of points (x,y) where f(x)-y=0. This zero-set definition of curves is more general, and if g(x,y) is a polynomial in x and y, then the set where g(x,y) = 0 is called an algebraic variety.  

In projective geometry, the Cayley-Bacharach Theorem states that for any two cubic curves in the plane, who intersect in 9 points, any third curve that passes through any 8 of those intersections, also intersects the ninth.

For this demonstration,  I've fixed 7 of the 9 points to be the unfilled dots O. If you move the red point, leaving the green point fixed, then all three curves will intersect at (0,1). If you move the green point, all three curves will change, but they will continue to intersect at some other point (which probably isn't (0,1)). If you move the green point, you can try to find the new intersection of all the points here.