How were 'old-school' mathematics graphics created?

Often the illustrations were drawn by hand, by the mathematicians themselves. The book A Topological Picturebook by George K. Francis (Springer, 1987) describes how one learns to do this:

This book is about how to draw mathematical pictures. … Theirs [the geometers of the 19th century] was a wonderfully straightforward way of looking at rather complex things, notably Riemann surfaces and geometrical constructions over the complex numbers. They drew pictures, built models and wrote manuals on how to do this. … I resolved to try to do the same for the mathematics of my contemporaries.

The first example is how to draw a hyperbolic paraboloid on the blackboard:

No software is used, but there are techniques one can learn.

Up until the 1990's, high-schools used to teach "technical drawing". Students would spend hours just drawing by hand isometric projections of various bits of machinery. To draw an ellipse, you would draw guidelines, a parallelogram, then several points of the circle, at various angle positions, 30, 45, 60 degrees, then sketch the curve and add lines.