Inscribing a circle in an Oblique Square (Drafting)

[DaveC426913]

This comes up in a drafting and illustration contexts. It's a mix of 2D and 3D geometry.

Since I was about twelve and first learning to draw mag wheels on racecars, I've been inscribing circles inside squares.

^{(Not mine. Stolen off Google)}

I noticed right away that it is not as simple as it might seem.
To replicate this, we start with a square, viewed obliquely (1).

Naively inscribing an ellipse into it with its axes aligned to the square's diagonal will result in the incorrect orientation that looks terrible (2).

To seat the circle correctly in the square, the circle must tangentially touch the centre points of all four sides of the square (3). This new shape is also an ellipse, but its major/minor axes are at an angle to the construction lines of the square.

My question is: is there a logic to the relationship between the ellipse's actual axes and the square's construction lines?

(There's other aspects to this question, such as:

• how does the relationship hold up when one and two-point perspective is added to the sketch?

• is this relationship related to the patterns on reflective disks?

but one thing at a time...)

 

[DaveC426913]

For reference, this is how we were taught to do it in High School Mechanical Drafting. Four circular arcs.

It's still a hack, but a more serviceable one.

 

[DaveC426913]

Not one taker, eh?

 

[pasmith]

I think the projection from $(x,y,z)$ space to the 2D $(X,Y)$ space of the paper is $$\begin{split} X &= x + y \cos \theta \\ Y &= z + y \sin \theta\end{split}$$ where $\theta$ is the angle between the horizontal ($X$) axis and the image of the $y$-axis. Then the image of the circle $(y - \frac12)^2 + (z - \frac12)^2 = 1$, $x = 0$ is $$(X\sec\theta - \tfrac12)^2 + (Y - X\tan\theta - \tfrac12)^2 = 1$$ and setting $(X,Y) = (u + \tfrac12\cos\theta, v + \tfrac12 (1 + \sin\theta))$ reduces this to $$u^2\sec^2 \theta + (v - u\tan\theta)^2 = \frac12$$ or $$\begin{pmatrix}u & v \end{pmatrix} \begin{pmatrix} 2(\sec^2\theta + \tan^2\theta) & - 2\tan\theta \\ - 2\tan\theta & 2 \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} = 1.$$ If $\lambda_1 \geq \lambda_2 > 0$ are the eigenvalues of this matrix then $1/\sqrt{\lambda_2}$ and $1/\sqrt{\lambda_1}$ are respectively the lengths of the semi-major and semi-minor axes, and the corresponding eigenvectors (which are orthogonal, since the matrix is symmetric) show how the axes of the ellipse are rotated relative to the $(u,v)$ axes (which are parallel to the $(X,Y)$ axes).

 

[Mark44]

Not one taker, eh?

OK, I'll bite.

For reference, this is how we were taught to do it in High School Mechanical Drafting. Four circular arcs.

I took a semester or maybe a year of mechanical drawing. As I recall, we used an ellipse template to create the illusion of a circle viewed at an angle.

Like this ... https://www.ebay.com/itm/2254658097...a45816c709bf9e38c7770e5fca8ae9a0&toolid=20006

 

[DaveC426913]

OK, I'll bite.
I took a semester or maybe a year of mechanical drawing. As I recall, we used an ellipse template to create the illusion of a circle viewed at an angle.

Like this ... https://www.ebay.com/itm/2254658097...a45816c709bf9e38c7770e5fca8ae9a0&toolid=20006

Yes. A template is certainly a way of making a true ellipse.

But note, it does not solve the problem of getting the angle right, which is the more primary focus of this thread.

In fact, arguably, it makes the problem potentially worse, because those templates invariably have the major/minor axes marked, and it's very tempting for a student to line the axes up to the diagonals of the square, as in figure 2. in the OP:

 

[DaveC426913]

.

They ... did not teach me about eigenvalues or matrices in high school...

 

[Mark44]

In fact, arguably, it makes the problem potentially worse, because those templates invariably have the major/minor axes marked, and it's very tempting for a student to line the axes up to the diagonals of the square, as in figure 2. in the OP:

The same student might be tempted to shove the pencil into his nose, as well, but that's not a valid argument against pencils or templates. As I recall, there are some templates with different shaped ellipses with some fatter and some thinner.

 

[Lnewqban]

Where the idea of “inscribing an ellipse into it with its axes aligned to the square's diagonal” comes from?

 

[DaveC426913]

The same student might be tempted to shove the pencil into his nose, as well, but that's not a valid argument against pencils or templates.

Granted.

I can't say why we were instructed how to make them using only a compass. Perhaps it had to do with working manually before using tools (like learning your times tables before using a calculator); perhaps they just didn't expect high school students taking a half-credit course to splurge on their own equipment.

Where the idea of “inscribing an ellipse into it with its axes aligned to the square's diagonal” comes from?

Perhaps I was surrounded by non-illustrators.