Lesson 1: Lines, Ellipses and Boxes

Ellipses are extremely important and notoriously annoying to draw. You'll find them all over the place in mechanical drawings. Cars, space ships, tanks, machines - anything man made will probably make extensive use of ellipses.

They also tend to feature prominently in organic drawings, especially as we look at the cross-sections of sausage-like forms.

Let's take some time to explore what they are, how they behave, and how they can be used.

For our purposes here in this course, we're going to be using ellipses in 2D space (drawn on the page) to represent circles that exist in 3D space.

Depending on whether you look at it as being two dimensional or three dimensional, this animated diagram depicts one of two things:

• In 2D space, it's a series of ellipses, from the widest to the narrowest.

• In 3D space, it's a circle being rotated, from facing towards the viewer, to facing away.

Keep in mind that since this is based on the angle between the direction the circle is facing and the direction the viewer is facing, the degree will actually change just by moving the circle around in space, without necessarily having to rotate it.

This diagram demonstrates how the further to the right a circle moves relative to the viewer, the more its face becomes visible, resulting in a wider ellipse representing it on the page, even without that circle rotating in space.

Throughout this course, we'll refer to this concept as the "degree shift" - basically the idea that the degree of our ellipse will shift wider or narrower, depending on how it moves in space.

We call the line that passes across the ellipse's narrowest span, and through its center, its minor axis. This minor axis line will cut the ellipse into two equal, symmetrical halves, so if it were to be cut out of the page and folded in half across this line, it would match up perfectly.

There's a major axis as well which runs perpendicular to it (across the widest span), but it's the minor axis that is actually the useful component.

Just as the 2D ellipse represents a 3D circle, the minor axis is extremely useful to us because of how that minor axis line coincides with and aligns with the normal vector - the 3D line that shoots straight off the surface of the 3D circle, completely perpendicular to that surface.

This means that we've got two ways of establishing the orientation of our circle in 3D space, as we draw it on our 2D page:

• The degree (width) of the ellipse controlling the circle's rotation

• The minor axis of the ellipse controlling the orientation of the circle's normal vector - basically the direction that the circle is facing.

If we need our circle to be oriented in a specific way in a more complicated scene, we will actually be able to start out with a normal vector line, then use it as the minor axis for our ellipse, drawing the ellipse around it so the line splits it into two equal, symmetrical halves, down its narrowest span.

Taking it further, the minor axis/normal vector is also extremely useful when we get into cylindrical forms. Here it serves as the "spine" to our form.

The ends of the cylinder share the same normal vector (in 3D) and minor axis (in 2D) - but more than just the ends, if you sample any cross-sectional slice along the length of the cylinder, it'll also align to the same minor axis/normal vector.

Fortunately for you, we don't need to worry about this right now - they won't come into play until Lesson 2. It's just useful to see how valuable the minor axis of an ellipse can be, specifically as we try to figure out all the ways in which we can represent different properties of a circle in 3D space.