Introduction to Perspective Drawing
A Brief History
Before introducing perspective into graphics, paints and drawing has no clear way to describe the relationship between objects in the frame. Different sizes according to the thematic importance seems to be the only option to establish the relations. However such paints cannot be called as a coherent picture, rather than a group of distinct symbols. Overlapping is one of the old method to show the distance relationship between objects, but it’s a poor architectural drawings where certain parts are not clearly shown. The following figure shows a nonperspective drawing.
Perspective first entered the artistic use around the 5th century B.C. in ancient Greece, where a flat panel is used to illustrate the concept of depth on the stage. The first set of geometric theories of perspective are developed by the Greek philosopher Anaxagoras and Democritus. Later Euclid’s Optics introduced a mathematical theory of perspectives. The most modern optical basis perspective was given in 1021 by Alhazen, who explained the light objects conically into the eyes. The first applicable geometric method of perspective drawing was found by Filippo Brunelleschi in 1415. He found that when painting the outline of building onto a mirror, the outline will converge on the horizon line by extending the outline far enough. The mathematical basis of perspective drawing is established during the period of Renaissance. During this time, a friend of Fillippo Brunelleschi, Leon Battista Alberti, showed not only the mathematics behind the conical projection, he also formulated the theory based on light rays from the viewer’s eyes to the landscape would strike the picture plane which is painting. The height of the object should appear on the drawing could be simply calculated with two similar triangles, illustrated in the following figure.
Types of Perspective Drawing
Before getting into the descriptions of different perspectives, we will need to define a few terms first. Vanishing point, VP, is where the parallel lines appears to converge in a perspective drawing. When we are look at a railway trail that extends far away in a straight line, the trails seems to converge into a single point at far distance, and this point is known as vanishing point. Horizon line is the horizontal line that cross the picture frame. It’s always at eye level and its placement determines how we are looking at the object, from a higher position or a lower position. In perspective drawings a virtual horizon line is always needed to define which part of the object will be revealed in the picture. Figure 4 presents a picture to illustrate both vanishing point and horizon line. By adding in them in the graphics, we are actively attempting to represent the reality with our drawings.There are several different ways to define graphical perspectives, namely linear, curvilinear and reverse perspectives. Each of them plays an important role in different types of graphical works. In normal artistic works, linear perspective with single or multiple vanishing points is often used. The distinction between different types of perspective is determined by the number and placement of the vanishing points used in the perspective technique. One to three vanishing points usually exists for linear perspective. Five vanishing points mapped into a circle with 4 VP’s at the cardinal heading of north, south, west and east directions and one in circle origin are used in curvilinear perspectives. Last the reverse perspective will have vanishing points placed
outside, “in front”, of the painting.
Linear Perspective
There are three sub-types of linear perspective, one-, two- and three-point perspective, and as discussed above, the points refers to number of VP presents in drawings. The following figure represents how would a cube being drawn with all three types of linear perspective. Each linear perspectives has its own advantage over the other one based on how the object shown in the picture plane is constructed. One-point perspective most fits when the objects are made up of lines either parallel or perpendicular to viewer’s light of sight. The scene of painting is composed of entirely linear elements that interacts at right angle, which means the painting plane need to be parallel with two of the scene’s axis. Two-point perspective can be used to draw same objects as one-points, however the objects will be shown as rotated in the picture. The two-point perspective is used when the panting plane is only parallel to one of the scene’s axis, z-axis in the above figure. Three-point perspective is usually used for building seen from above (or below) and the painting plane is not parallel to any of the scene’s axis. It’s like to look up a tall building where the third VP will be high in the sky.
There are three sub-types of linear perspective, one-, two- and three-point perspective, and as discussed above, the points refers to number of VP presents in drawings. The following figure represents how would a cube being drawn with all three types of linear perspective. Each linear perspectives has its own advantage over the other one based on how the object shown in the picture plane is constructed. One-point perspective most fits when the objects are made up of lines either parallel or perpendicular to viewer’s light of sight. The scene of painting is composed of entirely linear elements that interacts at right angle, which means the painting plane need to be parallel with two of the scene’s axis. Two-point perspective can be used to draw same objects as one-points, however the objects will be shown as rotated in the picture. The two-point perspective is used when the panting plane is only parallel to one of the scene’s axis, z-axis in the above figure. Three-point perspective is usually used for building seen from above (or below) and the painting plane is not parallel to any of the scene’s axis. It’s like to look up a tall building where the third VP will be high in the sky.
Next, let’s take a look at curvilinear perspective which an graphical project of 3D object on 2SD surface. The image of any real object will be projected onto the retina of eye as we are looking at them. A curvilinear perspective uses 5 VP to simulate the spherical projection shape of retina, and it gives a more accurate representation of how we are actually looking at any 3D objects. Following figure shows the effect of curvilinear project.
Reverse Perspective
Reverse perspective is another new graphical perspective that developed in the modern time. It’s also known as inverse perspective or Byzantine perspective. As we mention previously, the perspectives add in the depth into the pictures. Normally in linear and curvilinear perspective, the further the object, the smaller they appear to be. In contrast, reverse perspective will have the opposite effect. The vanishing point is place out of the picture plane so that the lines of the object will diverge against the horizon line.
Mathematics Basis
The perspectives in drawing are not determined randomly, they all follow a particular geometrical rules during the construction. Euclid’s work of similar triangles are first used to formulate the theory behind perspective projections. Here we shall focus on 2 core components of graphic perspectives.
Perspective Projection / Transformation
Any planar artistic drawings are a 2D representation of 3D object in the real world. When translate the 3D object on to planar frame, we are performing an projection of the object. There are geometrical principles that govern this transformation in order to achieve a precise perspective drawing. The projection could be thought as looking through the viewfinder of a camera. The position of the camera, the orientation, and the field of view will determine the behaviour of the transformation. The location of the camera is also known as the center of perspectivity. In order to find the matrix representation, we will separate the projection into two steps, one is to transform the object`s coordinate into the coordinates of the camera, the other one is the projection of the object onto the image space. Let’s define the following terms to illustrate the first transformation: ax,y,x is the point on the 3D object, cx,y,z is the location of the camera, Ѳx,y,z is the orientation of the camera, and ex,y,z is the viewer`s position relative to the picture plane. bx,y will be the 2D
projection of point a. The in the left-hand system, the new coordinate of point a, could be found as following. dx,y,z will be the transformation of a into the coordinate system define by c.
Using similar triangle, the 2D project onto the image plane could be found as
Then, using homogeneous coordinates, then the overall matrix representation will be.
And bx = fx/fw and by = fy/fw.
Thus this provides us a matrix transformation of any points on the 3D object. We could then
construct the 2D plot of the object by projecting all points on 3D object.
Desargues’ Two Triangle Theorem
In the perspective geometry , the Desargues’ Theorem states that, “ in a projective space, two triangles are in perspective axially if and only if they are in perspective centrally.” In another words, if the triangles have corresponding vertices jointed by the concurrent lines to form the perspective center, then the intersection of corresponding sides must be collinear and form the perspective axis. The converse must also be true, that is if two triangles’ corresponding sides are collinear, then their corresponding vertices are concurrent. The proof of this theorem is as following.
Considering the following figure the Desargues’ Theorem means that if PP’, QQ' RR’ al pass through one point O, perspective center, then the intersection points A by QR and Q' R', B by RP and R’P’ and C by PQ and P’Q all lies on one line.
We need to show that point C lies on the line AB. Suppose we have two perpendicular planes, the x-y plane and y-z plane, we then project the point P onto [-1 0 1]. Let two triangle PQR and P’Q’R’ lies in the y-z plane positioned so that line AB lies on the horizon line [x =0 z =1]. Since lines whose intersection is on the horizon line are parallel lines before projections, therefore by projection onto the x-y plane we obtain two triangles having two pairs of parallel sides. The remaining side must be parallel by similar triangles. Because the sides PQ and P’Q’ of the original triangles project into these parallel lines, their point of intersection C must lies on the vanishing line AB. The converse could be proved as following, suppose the triangle PQR and P’Q’R’ whose corresponding sides intersect in the three collinear points A,B,C and we need to prove that line RR’ coincident the intersection point Q created by line PP’ and QQ’. Here we shall apply the forward theorem to triangle AQQ’ and BPP’ whose joins of corresponding vertices all pass through C, while their intersections of corresponding sides are O R’ and R.
Conclusion
Perspective graphics adds realism into all kind of drawings. By studying them in depth, we could get a better understanding on how 3D objects appears to us and how should we to recreate them on the 2D plane more precisely. The different perspectives used in artistic works also based on geometrical and mathematical background. With the mathematical understanding of the perspective projection, we could write computer programs that will perform the necessary tasks for us. The perspective drawing also had a profound impact on the computer aided designs in the profession of architecture and engineering.
