Drawing
Drawing here begins with perspective: why far things are drawn small, where the horizon sits, and how vanishing points place every edge of a box.
Seven units around one pane of glass
The subject has one course so far, Perspective drawing, in seven units. The first sets up the picture plane, the second the horizon line, the next three cover one, two and three point perspective, the sixth measures equal spaces in depth, and the last tells the history from Brunelleschi to Brook Taylor. Every unit uses the same 3D model: a room with a tiled floor and three boxes, a framed sheet of glass, and a small dark sphere for the eye. Sliders move the eye height, turn the room and tilt the glass, and a flat drawing underneath shows the result on paper.
Similar triangles and the eye level line
Alberti wrote in 1435 that the picture works as an open window. Every point you draw is where a straight sight line from one fixed eye crosses that glass, so the drawing is correct from one viewing point only. Two similar triangles give the size rule: an object of height H at distance D, drawn on glass a distance d from the eye, comes out H × d ÷ D tall. A door 2 m tall and 6 m away, on glass 1.5 m from the eye, is drawn 0.5 m tall; double the distance and it halves. The horizon line is always at eye level. Below it you see the tops of things, above it their undersides, and it crosses every upright object on the floor at the height of your eye.
How many points the edges run to
A box has edges in three directions. Set it square to the glass and only the depth edges converge, to one point straight ahead on the horizon. Brook Taylor defined that point in 1719 as the place where a line through the eye, parallel to the edges, meets the picture plane. Turn the box by θ and the width edges converge as well: V1 lies d cot θ to one side and V2 d tan θ to the other, the lines from the eye to them meet at 90°, and their gap is never less than 2d. Tilt the glass up by τ and the upright edges join them at a third point, d ÷ tan τ above the centre, while the horizon drops d tan τ below it.
A hole, a mirror and a tiled floor
Equal spaces in the room do not stay equal on paper, so the sixth unit uses diagonals instead of a ruler: two diagonals find the true middle of a wall, and a line through the middle of one fence post places the next. The last unit starts with Giotto, whose parallel lines never met at one point. Brunelleschi, about 1413 to 1420, bored a hole at the vanishing point of his Baptistery panel and had viewers compare it with the real building in a mirror. Masaccio used the method in his Trinity fresco in the 1420s, Alberti wrote it down in Latin in 1435 and in Italian in 1436, and Desargues turned it into the start of projective geometry by 1639.
Questions about this subject
Why do parallel railway tracks seem to meet in a drawing?
Their real separation stays fixed, but the drawn gap becomes smaller as distance from the viewer increases. Perspective represents each point where its sight line crosses an imaginary picture plane. The point approached by the receding rails is a vanishing point, not a place where the actual tracks have physically bent towards one another. Perspective
Is the horizon line where the floor meets the back wall?
No: on an upright picture plane, the horizon is set by the viewer’s eye height. The join between floor and back wall can lie below it and does not define it. Raising the eye raises the horizon and changes which surfaces are visible, so a room still has a horizon even when walls hide any distant outdoor view. Perspective
When should I use two-point perspective?
Use it when two sets of horizontal edges recede at angles to an upright picture plane, as when viewing a building’s corner. Each set converges to its own vanishing point. The vertical edges remain parallel to the upright plane and stay vertical, so two-point perspective does not mean making every line converge towards one of two arbitrary dots. Perspective
Why do vertical buildings lean inward in some drawings and photographs?
Looking up can tilt the picture plane, so the building’s vertical edges are no longer parallel to that plane. They then converge just as the width and depth edges do, creating a third vanishing point. A phone camera pointed upwards follows the same geometry; keeping it level preserves parallel verticals but may cut off the building’s top. Perspective
How can I find the centre of a wall drawn in perspective?
Draw its two diagonals and use their intersection, rather than measuring halfway across its drawn width. The far half looks shorter than the near half, so the midpoint measured on paper is misleading. A vertical through the diagonal crossing divides the real wall into equal halves, and repeating the construction can locate quarters and eighths. Perspective