handprint : construction methods     perspective construction methods This page explains some of the most common geometrical construction methods or drafting tricks used in perspective drawing.   Bisecting an Angle. In many situations it is quicker to identify the measure points in 2PP by bisecting the angles between the dv and the left or right vanishing points. bisecting an angle   Our goal is to construct a line passing through point b so that it bisects the angle abc. There are three steps required, using a ruler, pencil and compass: 1. Draw an arc with the center at b and crossing both lines ab and cb. The intersections between the arc and these two lines defines two points, 1 and 2. 2. Draw two more arcs of equal radius with a center at 1 and 2, so that they cross each other within the angle abc. The intersection defines a new point, x. 3. Draw a line from b through x. This line bisects the angle abc.   Bisecting a Line. For some perspective constructions, such as the circle of Thales, it is necessary to find the midpoint of a line segment — that is, a section of line between two points. bisecting a line segment   Our goal is to find the point M that divides the line segment AB into two equal parts. There are two steps required, using a ruler, pencil and compass: 1. Draw an arc with its center at A and crossing the line AB at more than half the distance to B. Draw a second arc with its center at B, so that the two arcs intersect, at points x and y. The two arcs must have exactly the same radius. 2. Draw a line between points x and y. This line bisects the line segment AB at point M. (Note: the line xy is also perpendicular to AB.)   Constructing a Perpendicular. Many perspective construction methods require you to draw one line perpendicular to another. This is simple if you have a protractor or an architect's square, but you can also solve the problem, once again, using only a compass and straight edge (ruler). constructing a perpendicular line   Our goal is to construct a perpendicular line at point P on an existing line. There are just three steps required, using a ruler, pencil and compass: 1. Draw a circle with a radius of several inches and with the point P as its center (Cp, red circle). This circle will cross the existing line at two points, A1 and A2. 2. From each of the A crossing points, draw two more circles an inch or two larger than the first (Ca, blue circles). These will cross on opposite sides of the existing line, and each crossing defines a new point (NP). 3. Draw the new line (blue) through these two new points, or through one new point and the starting point P. The new line will be perpendicular to the existing line, and P will be at the corner of a right angle. What if you have to start with a point off the existing line, such as NP? No sweat: first draw a circle with NP as its center so that it crosses the existing line at two points, A1 and A2. Then proceed with steps 2 and 3, as before.   Equal Divisions of a Line. The most common construction task with perspective transversals or measure bars is dividing them into equal intervals. This is very simple to do with the equal intervals marked off on a ruler or yardstick. equal divisions of a line   The method is simply to draw two parallel lines from the ends of the line you want to divide, then find a ruler scale that will divide this width into the required intervals. The trick is that you can turn the ruler at any angle to the given line in order to make the intervals fit. In the figure, the given line is AB, and we want to divide this line into 3 equal intervals. We use the 1" scale on the ruler to give the required intervals, but unfortunately they given line is much shorter than 3 inches. No matter: we turn the ruler until the ends of the 3" length are exactly positioned over the two parallel lines at a and b. We then mark off the equal intervals at c and d, and carry these back to the given line with two more parallel lines. The intersection of these parallel lines divides the given line into thirds.   Proportional Lengths of a Line (Scaling a Line or Measurement Intervals to a New Length). In many situations it is necessary to find a length that is proportional to a given line. For example, you may need to extend a line to 160%, or cut its length to 70%. Usually you have a given length in perspective that represents a known measure (width, height, etc.) and you want to find a new length from that. The proportion you need is: proportion = new length in perspective/given length in perspective For example, if you have already drawn a line length in perspective that is equal to 150cm, and you require a length equal to 240cm, then the proportion is 1.6 (240/150). proportional lengths of a line   You use the same method as you would to divide the line into equal intervals (above), except now you must find two intervals on the ruler that represent the exact measures you need. There are two ways: 1. Find a ruler scale that gives you both measures in a suitable spacing. Look at all six scales on the ruler for a scale that is marked in 240 and 150, or some simple multiple of the desired intervals (24 and 15, or 48 and 30, etc.), where the interval 150 (15, 30) is slightly longer than the given line. You then can mark off the proportions directly. 2. Find a ruler scale that gives exact intervals for the given line only. The other interval will be found by multiplying this interval by the required proportion. Then, as before, draw two parallel lines from each end of the given line, lay the ruler across these lines until the required interval fits exactly between them, then mark off the new interval. Extend this back to the given line by a third parallel line. In the figure, the given line is AB. We extend two parallel lines from each end. We find a scale that divides this width into appropriate intervals (in the figure, the 1cm scale gives 5 major increments, 10 half increments and 50 minor increments). Then we lay the ruler across the two parallel lines so that they are exactly spanned by an even interval (5cm). To extend the line by 60%, multiply 5cm by 1.6 (8cm). Count off 3cm from the end of the line, and mark this 8cm length. Carry this back to the given line with a third parallel line, and extend the given line to meet it. The intersection marks the new endpoint of the line. To shorten the line to 70%, multiply 5cm by 0.7 (3.5cm). Count off 3.5cm from the end of the line, and mark this length. Carry this back to the given line with a third parallel line. The intersection marks the proportional cut. Last revised 08.01.2005 • © 2005 Bruce MacEvoy