# Why can ellipses be used to find squares in perspective drawing?

In Scott Robertson's How to Draw book he proposes a method for finding proportional squares using ellipses: (the "conditions" he mentions are described on the previous page and are simply that the left and right sides of the ellipse should meet the sides of the plane at their midpoints and the top and bottom points should be vertically aligned)

I understand that a circle always becomes an ellipse in perspective, but I don't understand the idea being proposed here that an ellipse fit to the side of a plane in perspective must be a perfect circle in plan view. What is to stop this ellipse being drawn here from also being a (differently proportioned) ellipse in plan view?

Through experimentation I have found this to be true though. Below are two perfect cubes in a 3D modelling program, and when overlaying the ellipse tool over both of them the only way to get the ellipse to fit in the way described by Scott ends up with the ellipse fitting perfectly on the face of one of the cubes. However, I have also found one edge case where this isn't true. When the ellipse minor axis is parallel to the horizon line any sized ellipse fits within the rules of what should create a perfect square.

By Scott's rules this is a perfect square: And yet so is this: So since this rule does indeed hold true in some cases, but not in all, my questions are:

1) What is the reasoning behind this rule? How does drawing an ellipse in this way result in a perfect circle?

2) When can this rule be applied, and when does it break down?

• The shown book spread text assumes in the beginning that one can already draw a single ellipse and a perspective square around it. (=draw at first an ellipse, then draw the one and only possible perspective square around it) The match between the square and ellipse is created with rules which are told in previous pages. I guess you do not follow those rules. I haven't that book, so writing a properly argumented answer instead of guesses isn't possible. But you will get one which assumes the book is perfect. Dec 22, 2019 at 12:58
• @JShorthouse Actually you can not do this. It is an often used approximation. But a circle will not become a oval in a perspective drawing. Also the major axes of said approximate ellipse are not along the axis you have drawn Dec 22, 2019 at 16:33
• Anyway the problem is that you have drawn the elliplse wrong. The major and minor axis do not always coincide the way you think. Anyway yo may want to look up the cord method. Dec 22, 2019 at 17:56
• @joojaa So a circle does indeed always become an ellipse, unless obviously if the circle doesn't fit entirely into the image plane? What was the point in being so pedantic about this? But anyway what do you think is wrong with my axes? The minor axis is always points towards the opposite vanishing point, right? Which ellipse in my examples is incorrectly drawn? Dec 22, 2019 at 20:51
• No, it does not hit all the edges tangentially. But its closer Dec 23, 2019 at 6:05

You're right that three tangent lines don't uniquely determine an ellipse. Given the minor axis as well, though, we have enough information. The point of Scott Robertson's method is to assume that the ellipse in question corresponds to a circle, and that the object is close to the centre of vision. Under those assumptions, the minor axis will effectively lie along the normal to the plane of the circle and passing through its centre. Thus, given three lines through a perspective square as well as its orientation, we can inscribe an ellipse to get a good sense of where to put the fourth side.

There are a couple of problems with Robertson's technique. For one, it falls apart if your object is not near the centre of vision. (He presents his technique as though it were universally true, but it is really only an approximation.) Consider the following cubes, for example: Since the cubes are distorted by the extreme perspective, Robertson's method would lead to a wildly incorrect drawing. The surface normal is not even close to the minor axis of the ellipse.

Another problem is the issue you raise where the minor axis lies along the horizon. In this case, it isn't that Robertson's method is wrong so much as insufficient. Consider the following cube: Ellipses inscribed on the two visible faces would have the same minor axis (namely, along the horizon), yet their degrees would obviously be very different. So you are right when you say "By Scott's rules this is a perfect square [...] and yet so is this." We essentially already need to know where the fourth side is (which we could judge from intuition based on the angle, or measure more accurately another way), which, of course, renders the ellipse method pointless.

In short, take Robertson's techniques as approximations that are useful in some, but not all, situations, and careful not to believe everything he claims.

You interpret terms "vertically aligned" and "midpoint" wrongly. They shouldn't be thought in terms of your drawing but in the imaginary view where a side and circle inside are straight on the face of the observer. Here's one side of the grid corner. Vanishing point V and points A and B are selected for good looks only. AB is one edge of a cube. AV and BV are the directions of 2 more edges. But placing the 4th edge GH can be made with an ellipse.

One must place the ellipse so that it has tangents AB, AV and BV. Only one ellipse will fit, there's no other choices. You have placed totally different ellipses, the 3 x tangency rule isn't respected at all.

The horizontal diagonal of the straight on the face circle is mapped onto a line which is between the tangency point C and V. C is the half height point on AB.

The half height point at GH is the crossing F. The tangent at F is the missing GH.

The main axes of the ellipse generally are NOT the perspective images of the horizontal and vertical diagonals of the straight on the face circle.

ADD: Finding G and H doesn't need drawing a tangent through F. As well one can draw lines from A and B to the crossing J. Extending those lines to cross AV and BV gives G and H The ellipse is still needed because J is the crossing of CV and DE. Without the ellipse you do not have C,D nor E.

• So just the clarify, the key here (which the book is missing) is that at the midpoints the tangents of the ellipse should have equal angles to edges of the plane? That makes a lot of sense, and while I knew before that my second drawing was obviously wrong it's now easy to prove that it is - the tangents where the ellipse meets the edges would basically be completely horizontal, they don't match the plane edges at all. Dec 22, 2019 at 21:10
• @JShorthouse No the book says this in the text for image 2. Its just slightly opaque. The example is bad though it is too straight so that you end up thinking of searching tangents in the wrong place. Dec 22, 2019 at 21:24
• @JShorthouse tangent of a curve is a line which meets the curve exactly in one point. Use that criteria. All angles are consequences of that fact, do not try to guess their relations. The images of the points where the circle inside the straight on the face square meet the square are C, D, E and F. They are the images of the the midpoints of the edges of the square. C,D,E and F differ from the midpoints of AB, AG, GH and BH. Dec 22, 2019 at 21:33
• @user287001 I'm not exactly sure what you're trying to say, are you just saying that the midpoints on the image plane are not the same as the actual midpoints of the square? Because I knew that already, sorry if I have been unclear. This is what I understand from your answer, is this correct? "The places where the ellipse meets the edges of the plane should be at the midpoints of the actual square, and at those points the tangent of the ellipse should be equal to the edge of the plane". Dec 23, 2019 at 16:17
• This is ok: "The places where the ellipse meets the edges of the plane should be at the midpoints of the actual square". The next is ambiquous: "at those points the tangent of the ellipse should be equal to the edge of the plane" Word "equal" =? Equal with what measures? "sameness" is an acceptable meaning for equal. Dec 23, 2019 at 17:21