A Guide to Drawing Spheres
When drawing a sphere turned in space, it can be difficult or tedious to accurately determine the relationships between the x, y, and z axes, and their respective ellipses. Here, I hope to show an approach that strikes a good balance between simplicity and accuracy. It is based upon three principles:
- Each ellipse will share its minor axis with a dimensional axis (x, y, or z).
- The ellipses will intersect at angles perpendicular to one another.
- The dimensional axes are also perpendicular to one another.
Because the intersections of the ellipses and the dimensional axes are both sets of three perpendicular lines, we can treat them as being identical.
This is combined with the observation that the minor axis of an ellipse will also be the same as a dimensional axis. With this knowledge, every line we place will allow us to deduce the other two. Don't worry about the technical details of this for now, just keep in mind that these sets of lines are related. Look for these relationships as you work through the diagram below.
To keep things from getting too complex, this method is demonstrated in an orthographic view.
Draw a circle and its center point.
Draw a line for the y-axis through the center. It can be at any angle you like.
The x-ellipse will share it's minor axis with the y-axis, so the major axis must be drawn perpendicular to it. The major axis will touch the edges of the circle, but the minor axis can be any length you want.
Place the x-ellipse.
Note: You can now interact with the diagram.
Draw the x-axis through the center of the x-ellipse, at any angle. The rest of the sphere can now be deduced.
Draw two lines parallel to the x-axis, tangent to either side of the sphere.
Draw a line connecting where the tangent lines touch the ellipse. This is the z-axis.
This technique is based on the geometry of how a circle fits inside of a square. Two lines parallel to a center line, if drawn tangent to the edges of the circle, will touch the circle at points directly opposite one another. Connecting those two points will give you a line perpendicular to the center line.
This also works in perspective, but rather than being parallel, the tangent lines must point towards the same vanishing point as the center line.
Next are the axes for the z-ellipse. The minor axis will be identical to the x-axis, and the major axis must be perpendicular to it.
The z-ellipse will touch the edges of the circle on its major axis, and cross the points where the z-axis meets the x-ellipse. Remember, the z-axis is the line where the x and z ellipses intersect.
There is only one possible ellipse that can satisfy these constraints.^{*}
^{*} You may run into trouble if the sphere is at an angle where the y-ellipse is seen directly from the side. In this circumstance, the major axis of the z-ellipse will overlap the x-axis, and the two sets of points will be identical. If you need to draw a sphere at an angle like this, you can transfer lines from a side view in order to determine how wide the visible ellipses will be.
The y-axis can now be shortened to where it meets the z-ellipse. This is the line where the y and z ellipses will intersect.
For the y-ellipse, the minor axis will share the z-axis, and the major axis will be perpendicular to it.
The y-ellipse will touch the edges of the circle on its major axis. The x and y axes mark its intersections, so its border needs to cross the points where the y-axis meets the z-ellipse, and where the x-axis meets the x-ellipse.
Keep the minor axis in mind as you determine the degree at which the y-ellipse must be drawn.
Finished. Although it may seem complicated at first, you'll soon start to intuitively understand the relationships between the ellipses, dimensional axes, and intersections. When you're comfortable with this method, try it without drawing all the construction lines. You can always add them in afterwards to see how you did.