How to Inscribe a Sphere into a Cube

How to Inscribe a Sphere into a Cube

Video by Vladimir London, a Life Drawing Academy tutor

In this video, you will see a very special method of drawing a cube, which I developed for Life Drawing Academy Correspondence Course students as well as how to inscribe a sphere into a cube. Many artists have troubles with drawing in perspective especially when the horizon line and vanishing points are outside the drawing area. Because such points are virtual, you do not need to plot them. There is a way to draw in perspective by using angles of vanishing lines. You need to make sure such lines are converging.

Let's begin with the horizontal line divided in half. Two vanishing lines go through its ends. These lines are converging towards the horizon. In the middle, there is one more line that follows this rule. Through the center, goes another line that points in the vanishing direction on the left hand side. It crosses the lateral lines in two points, marked here in green. A vertical line goes through the center. For this geometry, there is only one possible ellipse that can be plotted in such a way that it touches two green points. This means that for the given perspective, an ellipse cannot be fuller or slimmer. It is a perfect circle in this two-point perspective. Two more vanishing lines complete a perfect square in which this circle is inscribed. We know it is a square because the ellipse touches its four sides in the middle of each side. The elliptical oval gives the dimensions of a sphere. The outline of this sphere is a dark circle. One more axis of that sphere is perpendicular to the vanishing line that goes through its center. I mark this 90-degree angle. There is only one possible elliptical oval that fits this axis and crosses two green points. Note how I depicted the aerial perspective by making one half of this oval bolder than another half. Two green dots are marked on the equator of this sphere. I mark two more dots where this oval crosses the vertical axis. I outline the equator in a darker line that also follows the aerial perspective.

A perfect sphere is depicted in a given two-point perspective. At the top and bottom, I draw two vanishing lines through green dots. This sphere has the North Pole at the top, the South Pole at the bottom, and the equator in the middle, outlined as an elliptical oval. Through two green dots on the equator, I draw two vertical lines. They form a vertical square in perspective. Its corners are marked with small green dots. There is also a horizontal square in two-point perspective at the equator level. Its corners are marked in green. Through these green dots, I can draw vertical lines. Four straight lines complete the right-hand side vertical square. Four horizontal edges, connecting the left and right vertical squares complete the cube. I will outline its edges in darker lines. According to one of the main principles of constructive drawing, I draw all objects here as if they are transparent. This cube outlines also follow the rule of aerial perspective. Its back edges are slimmer than the frontal ones. So, here it is - a sphere is inscribed into a cube in a two-point perspective. It is a perfect cube because its diagonals on the bottom side are crossing in the South Pole and diagonals on the top side are crossing in the North Pole. Diagonal lines on the vertical sides are crossing in green points on the equator, where the sphere touches the cube.

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