Guide Programmed Learning in Perspective: A Guide to Program Writing

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A Guide to Program Writing I.K. Davies. Pogoloi Lolling Il Pooloolio A (B.A. Thomas GUIDE |.I. IEWICS D. UpoilsiläW TO I.B. BIT PROGRAM Orodri IV WRITING.
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On a ruler, this is generally the tick marked with the number zero. If we use two rulers, one perpendicular to the other, we can define the position of points in two dimensions. Add a third ruler, perpendicular to the first two, and you can define the position of points in three dimensions. The actual numbers representing the position of the point with respect to one of the tree rulers are called the points coordinates.

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We are all familiar with the concept of coordinates to mark were we are with respect to some reference point or line for example the Greenwich meridian. We can now define points in three dimensions. Let's imagine that you just bought a computer. This computer probably came in a box, and this box has eight corners sorry for stating the obvious. One way of describing this box, is to measure the distance of these 8 corners with respect to one of the corners. This corner acts as the origin of our coordinate system and obviously the distance of this reference corner with respect to itself will be 0 in all dimensions.

However the distance from the reference corner to the other seven corners, will be different than 0. Let's image that our box has the following dimensions:. Figure 3: a box can be described by specifying the coordinates of its eight corners in a Cartesian coordinate system. The first number represent the width, the second number the height, and the third number the corner's depth. Corner 1 as you can see, is the origin from which all the over corners have been measured. All you need to do from here, is somehow write a program in which you will define the concept of a three-dimensional point, and use it to store the coordinates of the eight points you just measured.

Like in any language, there is always different ways of doing the same thing. You have somehow created your first 3D program. It doesn't produce an image yet, but you can already store the description of a 3D object in memory.

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In CG, the collection of these objects is called a scene a scene also includes the concept of camera and lights but we will talk about this another time. As suggested before, we are lacking two very important things to make the process really complete and interesting. First to actually represent the box in the memory of the computer, ideally, we also need a system that defines how these eight points are connected to each other to make up the faces of the box. In CG, this is called the topology of the object an object is also called a model. We will talk about this in the Geometry section and the 3D Basic Render section in the lesson on rendering triangles and polygonal meshes.

Topology refers to how points which we generally call vertices are connected to each other to form faces or flat surfaces. These faces are also called polygons. The box would be made of six faces or six polygons and the set of polygons forms what we call a polygonal mesh or simply a mesh.

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The second thing we are missing, is a system to create an image of that box. This requires to actually project the corners of the box onto an imaginary canvas, a process we call perspective projection. Figure 4: if you connect the corners of the canvas to the eye which by default is aligned with our Cartesian coordinate system, and extend the lines further away into the scene, you get some sort of pyramid which we call a viewing frustum. Any object contained within the frustum or overlapping it is visible and will show up on the image. The process of projecting 3D point of the surface of the canvas, actually involves a special matrix called the perspective matrix don't worry if you don't know what a matrix is.

Using this matrix to project point is not absolutely necessary but makes things much easier. However, you don't really need mathematics and matrices to figure out how it works. You can see an image, or a canvas as some sort of flat surface placed some distance away from the eye. Trace four lines all starting from the eye to each one of the four corners of the canvas and extend these lines further away into the world as far as you can see.

You get a pyramid which we call a viewing frustum and not frustrum. The viewing frustum defines some sort of volume in 3D space and the canvas itself it just a plane cutting of this volume perpendicular to the eye line of sight. Place your box in front of the canvas. Next, trace a line from each corner of the box to the eye and mark a dot where the line intersects the canvas. Find out on the canvas, the dots corresponding to each of the twelve edges of the box, and trace a line between these dots.

What do you see?

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An image of the box. Figure 5: the box is move in front of our camera setup.


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The coordinates of the box corners are expressed with respect to this Cartesian coordinate system. Figure 7: the intersection points between these lines and the canvas are the projection of the box corners onto the canvas. By connecting these points to each other, an wireframe image of the box is created. The three rulers used to measure the coordinates of the box corner form what we call a coordinate system. It's a system in which points can be measured to.

All points' coordinates relate to this coordinate system. Note that a coordinate can either be positive or negative or zero depending on whether it's located on the right or the left of the ruler's origin the value 0.


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    7. In CG, this coordinate system is often called the world coordinate system, and the point 0,0,0 , the origin. Let's move the apex of the viewing frustum at the origin and orient the line of sight the view direction along the negative z-axis figure 3. Many graphics applications use this configuration as their default "viewing system". Keep in mind that the top of the pyramid is actually the point from which we will be looking at the scene. Let's also move the canvas one unit away from the origin.

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    Finally, let's move the box some distance away from the origin, so that it is fully contained within the volume of the frustum. Because the box is in a new position we moved it , the coordinates of its eight corners changed and we need to measure them again. Note that because the box is on left side of the ruler's origin from which we measure the object's depth, all depth coordinates which also called z-coordinates will be negative. Four of the corners are also below the reference point used to measure the object's height, and will have a negative height or y-coordinate.

    Finally, four of the corners will be to the left of the ruler's origin measuring the object's width: their width, or x-coordinates will also be negative. The new coordinates of the box's corners are:. Figure 8: the coordinates of the point P', the projection of P on the canvas can be computed using simple geometry. Let's look at our setup from the side and trace a line from one of the corners to the origin the viewpoint. As you can see these two triangles have the same origin A. They are also somehow copies of each other, in the sense that the angle defined by the edges AB and AC is the same as the angle defined by the edge AB', AC'.

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    Such triangles are said to be similar. Similar triangles have an interesting property: the ratio between their adjacent and opposite sides is the same. In other words:. Because the canvas is 1 unit away from the origin, we know that AB' equals 1. We also know the position of B and C which are the z depth and y coordinates height respectively of the corner.