Circles and lines – that sounded manageable. So, this was one of my first attempts at reviving descriptive geometry.

A circle on a sphere is projected into a circle in the equatorial plane under stereographic projection. But if the source circle contains the North Pole, the projection ray in this point becomes tangent to the sphere and parallel to the equatorial plane. Thus circles containing the North Pole are projected into lines – circles with infinite radii.

The drawing shows this projection scenario from three viewpoints: The “3D” view is constructed from the top view and the side view.

Geometrical construction / descriptive geometry / axonometry: Circles on a sphere containing the North Pole are projected onto lines via stereographic projection.
Stereographic Projection by elkement 2021: Geometrical construction / descriptive geometry / axonometry: Circles on a sphere containing the North Pole are projected onto lines via stereographic projection. Graphite pencil, notebook. Scanned, contrast enhanced, colored digitally.

I had just discovered digital contour plots of functions, and I wanted to color this drawing on paper accordingly. But four circles (ellipses) and their corresponding lines were all I could handle. So, this was not exactly colorful!

Fast forward a few years, and I discovered watercolor pencils! This was the safe way of using watercolors: Apply the pigment in a controlled way with the pencil, then timidly add the water with a brush!

At that time, I had been oscillating between enhancing the construction lines with a black or a white pen. In this case, I used both for the eponymous circles and lines in the view in the “3D” view:

Circles to Lines by elkement 2024. Drawing of stereographic projections of circles on a sphere onto lines in the equatorial plane, by elkement 2024. Traditional descriptive geometry, axonometry based on the projections from "above" and from "the side". Pencil, watercolor pencil, ballpoint pen, gel pen on watercolor paper.
Circles to Lines by elkement 2024: Seven circles on the surface of a sphere, projected onto seven lines in the equatorial plane. Only five of those lines are visible within the boundaries of the drawing surface. Graphite pencil, watercolor pencils, black, ballpoint pen, white gel pen on watercolor paper.

I still like this drawing. But I am not sure if I like the paper.

Watercolor paper is an awesome surface to paint on. It feels magic to let the water do its thing and create color gradients. But in the end, it is a piece of flexible and fragile paper. It requires framing or mounting.

I had tried different ways of gluing watercolor art to panels – from cardboard or from foam board. I created free-standing double-sided art. But there is nothing that feels like stretched canvas! Last year, I gessoed and sanded a 30cm x 30xm canvas: Constructing 3D Lissajous Figures on it and coloring them in my new stained-glass style ended up being the most involved process I had ever tried.

The smooth canvas is not absorbent, and the watercolor pigment does not stick to it. The upside: You can correct anything, indefinitely. The downside: You cannot really layer colors in the typical watercolor way – painting over the first layer when the color has dried, to create two translucent layers of color. With sanded canvas, the bottom layer will nearly be dissolved and lifted again. I had to be extremely careful and integrate this effect deliberately. I touched each of the little “cells” several times, careful mixing the new color with the existing one.

But there is another upside: The pigment stays on top of the surface – and it remains very vibrant, very unlike watercolor on paper!

So in 2026 I decided I want to try this again. To focus on the coloring technique rather than on the construction, I turned to my beloved Circles to Lines.

I am using gesso generously, and I found out that irregularities will be smoothed out after the canvas will have dried. Still wet, it looks messy:

Photo of a stretched canvas, freshly gessoed. In the background; An earlier version of this (to be) drawing, showing how circles on a sphere containing the North Pole are projected onto lines. By elkement 2024/2026.

I am doing the construction again: Creating the characteristic points of each circle / ellipse by intersecting the projection rays. (I am skipping the great circle visible in the 2024 version.)

Using a tiny bit of watercolor pencils to remember what belongs to what:

Photo of a geometrical pencil construction on a gessoed and sanded canvas: Circles on a sphere containing the North Pole are projected onto lines, constructed as an orhtographic projection based on the views from the top and the side. By elkement 2026.

While watercolor pigment can be moved around easily, graphite pencil dust sticks to the canvas. I need to clean up with eraser as much as this is possible at at!

Close-up of a photo of a geometrical pencil construction on a gessoed and sanded canvas, showing smudges: Circles on a sphere containing the North Pole are projected onto lines, constructed as an orhtographic projection based on the views from the top and the side. By elkement 2026.

Finally – I am ready for coloring! No more pencils: I am using real watercolors right from the start.

I cannot create calm gradients, due to the water-repellent nature of this surface. I pivot to smoky clouds!

Photo of a canvas with a geometric drawing, work in progress, traditional descriptive geometry. Started: coloring the tesselation generated by the projection rays with watercolors. Six circles on a sphere containing the North Pole are projected onto lines. By elkement 2026.

And again, I am touching each little cell several times, covering the canvas with the color in the most inefficient and slow way possible:

Photo of a canvas with a geometric drawing, work in progress, traditional descriptive geometry, coloring the tesselation generated by the projection rays with watercolors: Six circles on a sphere containing the North Pole are projected onto lines. By elkement 2026.

I am always aiming at a balance between creating a “scientific abstract painting” in the style of stained glass – and preserving ALL of the original construction.

Having practiced fine brush strokes, this is the artwork where I can finally use them!

Photo of a geometric drawing on canvas, work in progress, traditional descriptive geometry. The tesselation generated by the projection rays has been colored with watercolors. In progress: Re-tracing all the construction line with a fine watercolor brush. Six circles on a sphere containing the North Pole are projected onto lines. By elkement 2026.

The sides of the canvas frame should match the painting:

Photo of a geometric drawing on canvas, showing the painted edges. Traditional descriptive geometry. The tesselation generated by the projection rays has been colored with watercolors, and all the construction line shave been re-traced with a fine watercolor brush. Six circles on a sphere containing the North Pole are projected onto lines. By elkement 2026.
Photo of a geometric drawing on canvas, showing the painted edges. Traditional descriptive geometry. The tesselation generated by the projection rays has been colored with watercolors, and all the construction lines have been re-traced with a fine watercolor brush. Six circles on a sphere containing the North Pole are projected onto lines. By elkement 2026.

There is one, scary step left: As a few droplets of water could already damage this painting, varnishing is imperative. But I know from trial and error that there is a change that varnish may activate colors. This time, I do not want to turn this into a “mixed media” painting by using permanent markers to repair blurry lines. This has to stay watercolor only!

I am applying several faint layers of protection with fixative spray first, and then several layers of transparent glossy spray coating: Fortunately, no color changes and no bleeding!

This is the official photo:

Photo of a geometric drawing on canvas, final varnished version. Traditional descriptive geometry. The tesselation generated by the projection rays has been colored with watercolors, and all the construction line shave been re-traced with a fine watercolor brush. Six circles on a sphere containing the North Pole are projected onto lines. By elkement 2026.

Now I can release it to the world!

It might be a printed on canvas; here is a mockup:

Mockup of an art print on stretched canvas, created from a photo of a geometric drawing on canvas, final varnished version. Traditional descriptive geometry. The tesselation generated by the projection rays has been colored with watercolors, and all the construction line shave been re-traced with a fine watercolor brush. Six circles on a sphere containing the North Pole are projected onto lines. By elkement 2026.

To preserve the spirit of the painted edges, I have extended the painting by a mirrored and (manually, digitally) distorted version of it:

Mockup of a canvas print, printed from original mathematical art on canvas (stereographic projection, colorful watercolor). The original painting has been mirrored and distorted (digitally) on the edges. By elkement 2026.