Batch 3 - Class 206 - Orthogonal Projection and Perspectives

• Conceptualize some other interesting machines in Planiverse

• In a rectangular room with dimensions 30x12x12, a spider is located in the middle of one 12x12 wall one foot away from the ceiling. A fly is in the middle of the opposite wall one foot away from the floor. If the fly remains stationary, what is the shortest total distance the spider must crawl along the walls, ceiling, and floor in order to capture the fly? 
• The answer, 40, can be obtained by "flattening" the walls as illustrated below. Note that his distance is shorter than the 42 the spider would have to travel if first crawling along the wall to the floor, then across the floor, then up one foot to get to the fly.

• What if the room was 10x10x10
• In this case, flattening in the manner above "loses" a path. Its easier to just go across the ceiling and down. This path is visible in an alternate "flattening" of the room.

• Orthogonal Projections
• Explain the notion of orthogonal projection to kids, by taking a 3D object and drawing its three perspectives
• Let kids take some objects and draw their orthogonal projections
• Find all possible cubical buildings that have a base, and front projection and a right side projection that is as follows

• Answer: 7 solutions. 2 with 6 cubes, 4 with 7 cubes, 1 with 8 cubes
• Is there number of cubes determined by the following perspectives

• Answer: Yes

• Is there number of cubes determined by the following perspectives

• Answer: Yes

• Use exactly 20 cubes to make a model from the building plans below. Record the base plan for your building. What are the maximum and minimum numbers of cubes that could be used to build the structure. 

• Answer: The range of number of cubes is 18-22. 20 cube answer is as under (there are several answers)

• Explain the notion of perspective projection - parallel lines going away from the viewer seem to converge at infinite distance. Since we can't really model infinite distance, we can think of these as converging far away
• Consider a simple hut. Draw its orthogonal projection, and a perspective view

• Perspectives are evident in paintings (and in photos of course)

• One point projection - the whole scene is projected onto one plane (i.e. there is one point of convergence). Such as the box below

• If there is a sequence of 3-D boxes, we could draw one point projection as under

• What is wrong with the above? The face of these boxes is not really drawn correctly
• Draw a straight road vanishing in the distance using one-point projection. Draw line of trees on sides of the road
• Road in center
• Trees - farther ones are smaller and have more details. They are of course hidden partially by the nearer trees. These techniques help accentuate the perception of depth

• Two point projection - allows us to choose two planes of projection, and hence two points of convergence

• You are standing at the intersection of streets, with a building in front. Draw the perspective

• Think about three point perspective, and how you would see a tall building for example. What is the third dimension that could be subject to perspective?
• Three point projection - a box where eye is not aligned to the base of box

• Think of a tall building or tower or church, and draw a three point perspective