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2 Art and Geometry

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2 Art and Geometry 2 Art and Geometry 2.1 Perspective 2.1.1 What is Perspective? Have you ever looked at a painting of a pretty scene of the country side and wondered how the artist was able to capture the beauty and essence of the landscape in the picture. Or perhaps, you wondered how the artist was able to create a three dimensional effect on a piece paper of that is only one dimensional. All these effects can be created by using different types of perspective. Perspective is the art of drawing objects in manner that creates three dimensional effect or depth in the drawing. There are several ways to create a perspective in art. The artist can use light, shading, variance in the size of objects, and lines of sight to create depth perception in a drawing or painting. As it turns out, there are several types of perspective. In this course, we will study four types of perspective. Types of Perspective 1. Diminishing Sizes 2. Overlapping Shapes 3. Atmospheric Perspective 4. One Point Perspective As we study each type of perspective, we will look both art work and photographs that exhibit each type of perspective. 2.1.2 Diminishing Sizes In diminishing sizes depth is created by systematically making objects smaller. Objects that are closer to the eye are naturally larger that objects that are further from the eye. In some drawings or paintings, artist use variances in sizes to create depth perception. Diminishing sizes also are prevalent in photographs. In this first example, you can see how diminishing sizes occurs in a regular photograph. Notice that the first tower of the Golden Gate Bridge is much larger that the second tower. A depth perception is created naturally in the photo due to the different sizes of the objects in the photograph. If we look closer at the photo, we can also see that the suspension cables of the bridge get smaller as the bridge get further away. 58 Here are some examples of actual artwork that make use of diminishing sizes to create perspective. Example 1: Pieter Bruegal’s The Peasant Wedding Example 2: Pieter Bruegal’s Village Scene with Dance Around the May Pole 59 Summary of Diminishing Sizes 1. Diminishing sizes makes use of the position of the object and shading to create perspec- tive. 2. Diminishing sizes can be used with other perspectives to create perspective, in particular it can be used with on point perspective. 2.1.3 Overlapping Shapes In overlapping shapes depth perception is created by using overlapping shapes. Overlapping shapes create a sense of space and depth in a drawing. By placing one object behind another object, an artist can create the space of three dimensions to the flat two-dimensional space of the paper. When shapes overlap each other in a drawing, it creates an illusion called shallow space. This illusion help create depth in the drawing. We strengthen the illusion of objects going back in space by placing them one behind another along the page. Here is a painting by Rita Auerbach in titled Buffalo Waterfront Montage that exhibits perspective using overlapping shapes. 60 Notice how the artist creates perspective by overlapping the boats in the bottom of the painting and the buildings near the top of the building. In the next example, notice how overlapping shapes can occur in a normal photograph. In the photograph, the building naturally overlap each other creating a depth perception. Summary of Overlapping Shapes 1. Overlapping create illusion called shallow space by overlapping objects to create per- spective. 2. Overlapping shapes can be used with other perspectives to create perspective. 2.1.4 Atmospheric Perspective In atmospheric perspective depth is created by making objects that are farther away less clear by diminishing both color and shading. If you have ever looked at objects such as mountains at a distance, you will notice that they seem less vivid than objects that are closer. This effect can be captured in a painting or drawing by using different shading on 61 objects closer to the naked eye than object farther from the eye. Again, this perspective can also be clearly seen in a photograph as well. In next photograph, you can see how atmospheric perspective appears to the naked eye in real life. Notice that object closer the camera, the plants and buildings, appear to be much more vivid, where as objects further away from the camera, the mountains and San Francisco Bay Bridge, are less vivid. The next example shown below is a painting by Albert Bierstadt entitled ”Garden of the Rockies” Notice that the mountains and sky in the background that are farther from the eye are shaded differently than object that are closer to the eye. 62 Summary of Atmospheric Perspective 1. Atmospheric perspective uses color, shading, and vividness to create perspective. 2. Atmospheric perspective can be used with other perspectives to create perspective. 2.1.5 One Point Perspective During the Renaissance period many artists wanted to create drawings and painting that look more realistic. In particular artists had trouble painting figures such as building, bridges, streets, and other man made structures. In search for answer Filippo Brunellechi , an Italian architect, discovered a technique called one point perspective. This technique helped artists paint scenes of man made structures with a more realistic appearance. In one perspective depth is created by using lines of site that all connect to a point of focus called a vanishing point. The first Example of One point perspective we will examine is Vincent Van Gogh’s Flower Beds of Holland. In Van Gogh’s paints painting you can see how the rows of the flowers all meet at vanishing point. In the next illustration, you see how the rows of the flowers bed all converge to one vanishing point. 63 You can also see one point perspective in a actual photograph. Notice how we can see that the cable car tracks meet at a vanishing point in the photograph. Notice that the vanishing point is located at a point near the end of the street or below the tower of the San Francisco Bay Bridge. If you have ever walked on a straight stretch of railroad tracks, you usually can look off in the distance and image where the rails of the tracks would meet at a vanishing point. (See diagram below) 64 Summary of Atmospheric Perpective 1. One Perspective is the only one of the four perspectives we study that uses a focal point or vanishing point. 2. One point perspectives use a vanishing point along with line of site to create perspective. 3. One point perspective can be used with other perspectives to create perspective, espe- cially with diminishing sizes. 2.1.6 Dominant Perspective In many drawings there may be more than one perspective in effect in the drawing. For example a painting or drawing could have elements of one point perspective, diminishing sizes, and atmospheric perspective. If there is a perspective that stands out more than other perspectives, then that perspective would be the dominant perspective. In the next photograph, one point perspective would be the dominant perspective. Notice how the other perspectives, Diminishing sizes and overlapping shapes, are not as prevalent in the photograph as one point perspective is in the photograph. 65 Art without perspective Some paintings or drawings will not have a visible perspective in the picture. If this is the case, then the artwork will appear flat without any depth perception. Notice how the Egyptian hieroglyphics in the picture below have flat appearance with out any perspective. Often Egyptian hieroglyphics will have a lack of perspective. 2.1.7 Similar Triangles and Perspective Suppose we asked to make a drawing with a few trees in it and we were asked to draw the trees using diminishing sizes to create a perspective in the drawing. How could we use math to figure out how to space the trees apart from one another. The answer to the problem is to use similar triangles and proportions to find the correct distances. Before we work example like this, let’s review a few things about similar triangles. In the diagram below is a picture of two triangles that are similar. Since these triangles are similar, the sides of two similar triangles are proportional. Given that the sides of the triangle are proportional, we can set up the following ratio between the sides. ℎ1 = ℎ2 d1 d2 Now, let try using proportions to find the missing side of the triangle. Example 3 Given that the triangles below are proportional, find ℎ2 66 Solution Set up a proportion and solve for the missing variable which is ℎ2 ℎ1 = ℎ2 d1 d2 5 ℎ2 8 = 14 Now, find the cross-product of the proportion in the problem. 8ℎ2 = 5(14) 8ℎ2 = 70 ℎ2 = 8 .75 Here is an example of using a proportion to find the correct distance in drawing. Example 4 Use the picture below and the values of a,b,c,d, and e to find the missing value. If a =5 in, b = 3 in, and e = 12 in, find d. Round your answer to 2 decimal places. Solution: Draw a pair of similar triangles from the diagram above: 67 Set up a proportion and solve for d. a b e = d 5 3 12 = d Now, find the cross-product of the proportion in the problem. 5d = 12(3) 5d = 36 d = 7 .2in 68 Example 5 Use the picture below and the values of a,b,c,d, and e to find the missing value.
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