Equations involving two variables generally have graphs that correspond to curves. If we can solve for the dependent variable as a function of the independent variable, then the graph can be generated as a plot of a function. When we want to create a plot of the equation and do not solve the equation as a function, we create an implicit plot.
The equation of a circle with a center at \((x,y)=(2,-1)\) and a radius of \(r=3\) is given by \((x-2)^2+(y+1)^2=9\). This is not a function because it fails the vertical line test. We generate a graph below.
In Sage, the equals sign is used for different purposes. One equal sign is used to assign a value or a formula to a variable. Equations are a logical statement, which uses two equal signs.
Just for fun, here is another implicit plot that I think looks really cool. This is called a lemniscate, and it looks like the infinity symbol.
Be aware that the way that an implicit plot is drawn is that the computer evaluates the two sides of the equation at many points in the plane. It keeps track of whether the left hand side or the right hand side is greater. The curve is drawn at locations where the inequalities reverse. If the curve is too small, it may be that this process fails to find any points where the inequality reverses and nothing will be drawn.
