I am currently taking an architecture related art class in which we’re covering the basics of perspective drawing. In our last class period the teacher lectured on how to measure the depth of a cube with a method that I find egregiously and offensively erroneous and I would like some help in developing some kind of method to succinctly and absolutely discredit it.
Here is how the scene goes: The teacher draws a vertical line. This is the front edge of a cube. The edge closest to us as we sit on the station point. She then draws vanishing points on either side of the cube, which lie upon a common horizon line as any vanishing point must. Next come two lines from each end of the vertical edge, connecting them to each of the VPs. This gives us the planes of the two front faces of the cube.
The trick now is knowing how far back along each of those planes to place the rear edges of the cube. Actually, it is not a trick but an easily researched method and it involves the construction of measuring points to establish the forshortened scale along those receding axes.
But this is not what the teacher taught us. Instead, she insisted that for each side, one needs merely to bisect the angle formed by the leading edge and the lower receding line (meaning bisect the flat projection we see on paper) and drop the back edge from where that line intersects the upper receding line. The example she drew on the board was laughably non-cubular and I immediately protested the method, going so far as to create an example of a true cube with an X on the side in SketchUp, print out a 2-point perspective screenshot, and highlight the obviously dissimilar angles in question, illustrating my contention that the relationship of these angles to the depth of the form changes with the viewing angle.
I’m sorry that I can’t provide pictures for this as I know it may be difficult to understand what I’m talking about but if anyone has an idea for how to demonstrate to a stubborn teacher full of conviction that this is incorrect, please do share. Visual evidence and plain fucking common sense don’t seem to be enough.
You could look at a special case, where the top point of the vertical line is on the horizon, and the lines from the bottom point are 30 degrees from horizontal. Then you can use simple geometry to calculate how far to the left the left edge of the cube will be if you bisect the top angle, and how far to the left it will be if you bisect the bottom angle. If those distances are different, then the method fails.
It’s probably easier to figure out where the line is when bisecting the bottom angle, and then drop that line down and show that where that intersects the bottom line doesn’t lie on the (45 degree) angle bisector of the top angle.
That said, I tried it, and it did look close. Could this be a common approximation people use that looks close enough?
Regardless if it’s a fast trick or approximation, what should be taught are the fundamentals, then an instructor can get into tips and tricks for common scenarios. So, yes, she’s in err and ZenBeam’s method should demonstrate how and why such shortcuts are disaterous if the underlying principles of the geometry aren’t correctly taught first.
She sounds like the obstinate type of teacher that doesn’t like to hear she may be doing her students a disservice and/or simply doesn’t know herself the proper techniques in that instance.
That said, perhaps approach her one-on-one how you’re trying, and failing short, in using her method on solving depth in drawing a cube at off-center/non-45° angles, then ask her for her “help”?
Sure it’s disingenuous, but playing ignorant might defuse her stubbornness and open her up a bit.
Using ZenBeam’s idea, the math fails. I can’t construct an elegant proof right now. But using his method, the bisected angles form neat 30-60-90 and 45-45-90 right triangles.
Looking at just one side of the image, we assume the horizontal edge of the projected shape to be a units long. The outer edge of the cube formed by the 45-90 triangle would also be a units. The horizontal edge would form the short side of the 30-60-90 triangle, and its hypotenuse would be a diagonal line on this side of the cube at 2a. The triangle formed by the outer edge, the lower edge, and the diagonal would be 30-30-120, with side lengths of a, a, 2a - which would be impossible.
Drawing a cube to scale in perspective is as easy as drawing a scale overhead view of the cube (a square), a vantage point, and a horizon/view-plane at one of the cube vertices. Then, the height of the cube at that vertex will be at the same scale as the overhead plan-view, and all the other vertical edges can be referenced by extending lines from the vantage point. Put the vanishing points along the horizon line so they form a right triangle with vantage point, and the rest is 6th-grade-art-class level stuff.
It is a touch more complicated than drawing an arbitrary edge and bisecting some angles, but it is accurate. However, I suspect this class is much more about art than drafting, and the instructor has never taken a drafting course.
This is an “architecture related art class”, not geometry. I took drafting back in high school, and one thing I remember being taught, because it was “wrong”, was how to draw a circle in an isometric view. The correct shape is of course an ellipse, but that isn’t how we drew it. If you imagine a circle inscribed in a square, in the isometric view, the square becomes a rhombus, and the inscribed circle an ellipse. The ellipse was drawn as four circular arcs, two arcs with a small radius for the sharp corners of the rhombus, and two with larger radii for the obtuse corners. The arcs came together smoothly in the midpoints of the the sides of the rhombus. It looked fine.
Now the OP does say
so maybe there is one true way of doing this. But it’s also possible there’s a generally accepted shortcut that looks close enough.
Perhaps, I’ve taken plenty of drafting and art/illustration courses myself. A trick is still a trick and should be taught as such. One method may be superior over others in certain scenarios. To insist one technique, used as a shortcut, will be true in all cases, well… now your trying to shoehorn these tactics into geometric absolutes. A slippery slope to be teaching on, especially if the instructor isn’t aware of this.
