If you draw a square on a sheet of paper, and draw the diagonal, are the apparent angles from the diagonal to each side equal, even if you rotate the sheet in 3D space?
Transcript of Core Problem:
A parchment scroll with a drawing of a square on it. There is a red line defining a diagonal, and two angles. Theta one defines the angle between the diagonal and the top side, and theta two defines the angle between the diagonal and the left side.
Obviously, theta 1 equals theta 2. However…
Three scrolls with the same square on it, but each scroll is rotated in 3D space.
- The first scroll is rotated along the vertical axis and drawn in single point perspective.
- The second scroll is rotated on the horizontal axis and laid flat, then lightly rotated along the vertical axis, with its vanishing point resting just above it.
- The third scroll is rotated further on the Y axis than the second, and is drawn in two point perspective, the vanishing points far to the left and right of the scroll.
Does Theta one still equal theta two?
Transcript of Practical Application
A drawing of a cube(?) in two point perspective, with the left face drawn thinner due to foreshortening. A red diagonal bisects the left face, with the angles theta one and two defining the upper and lower angle. Theta one is clearly larger than theta two
The apparent size of theta one is greater than theta two. Does this mean I drew the cube wrong?
This question mainly applies to drawing squares in one, two and three point perspectives, and does not apply to four point or fish-eye perspective, since straight lines and angles stop really being a thing past that.
Related questions are “Does a bisected angle drawn on a sheet retain equal sub angles no matter the orientation of the sheet?”, and the parent question: “how do i draw a cube pls”
(Bonus scribbles of madness from trying to solve this problem)
Studying projective geometry is one of the most beautiful parts of elementary mathematics, and it's definitely what you're looking for
When you take an aerial photo of a city, for example, the angles change and the length ratios change, but despite that, when we take a photo and then take a photo of this photo and take a photo of the new photo and so on... we can still distinguish objects correctly. The reason for this is to preserve Certain properties in the original form, which are the properties that are not affected by projective transformations. In particular, the points on the same line in the original form remain on the same line in the form taken from them, and for four points on the same line the reciprocal ratio will remain constant. There are no distinct properties. It is kept for three points other than straightness. Even the arrangement will differ in the general case, but it will not be affected in the aerial photograph due to material considerations and nothing more.
In projective geometry, it will be useful for you to become familiar with some new terms, such as the vanishing point, which is the point of intersection of parallel lines, as it appears when looking at railways. On the other hand, you must recognize the point of intersection of parallel lines located at infinity.
Also, if you want to study projective geometry, it would be good for you to learn the concept of duality, which will enable you to convert points into lines and lines into points, and thus deduce new theories from other theories you know.
Perhaps it will also be useful for you to become familiar with some projective geometry theorems, such as Desargues's theorem, Pappus's theorem, and Pascal's theorem.
As for your question about preserving angles, this is incorrect. There are no angles that are preserved under projective transformations
But we can say that conic sections remain conic sections under projective transformations (and their type may change).
Here are some inspiring photos:
Finally, here are some useful links:
https://en.m.wikipedia.org/wiki/Projective_geometry
https://en.m.wikipedia.org/wiki/Cross-ratio
https://en.m.wikipedia.org/wiki/Duality_(projective_geometry)
https://en.m.wikipedia.org/wiki/Pascal%27s_theorem
https://en.m.wikipedia.org/wiki/Pappus%27s_hexagon_theorem
https://en.m.wikipedia.org/wiki/Desargues%27s_theorem
https://youtu.be/rMyF104x_VU?si=M9CttImYH_OZsb8S