CGRA 151: Introduction to Computer Graphics Assignment Sample NZ

Students typically think of computer graphics as a way to make movies and video games. Computer graphics technology has increased the quality and realism of both, but there is much more to it than that. The field includes fields such as industrial design, architectural rendering, medical imaging, scientific visualization, and many others. Computer Graphics students work with 3D modeling software like Maya or Blender to create everything from images for an architect’s drawings to animated sequences for film production.

Computer Graphics students at NZ learn many of the same things as other CG students like animation, rendering, and modeling. The difference is that they come away with a broader range of knowledge since their time is split between digital artwork and artistic projects. It doesn’t matter what type of CG you want to work in; it’s important that you take the time to create things by hand. Even if you don’t plan on being an artist, having an understanding of artistic concepts will help you in your CG work.

The course has four components:

1. Programming: To teach you how to use a Java-based graphics language, Processing, to consolidate what you learned in COMP 102, COMP 112, or DSDN 142. To teach you something about algorithm design, especially about ways to optimize an algorithm.
2. Behind the scenes: Detailed consideration of a number of fundamental computer graphics algorithms that allow you to understand what a graphics card does when it draws.
3. Underlying mathematics. Algebraic representations of lines and curves. Vectors, matrices, representation of transforms using matrices. Algebra for line-line intersection and closest-point-to-a-line calculation.
4. Fundamental concepts in human vision, color representation, and display design: so that you know the limitations of what we do and why those limitations exist.

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This course will increase your students’ knowledge and understanding of the subject. The following are some activities that will be answered in this course:

Assignment Activity 1: Write simple programs in the Processing programming language

Processing is a Java-based programming language and the environment particularly targeted at visual arts. It enables artists, designers, filmmakers, video game players to create computer animations with code.

Essentially this programming syntax was created for artists that are more interested in the end result of their work rather than tedious lines of code that often mimic other already existing programs or coding languages. Processing has yet to require any installation files which makes it an excellent beginner platform for anyone wishing to learn how to program computers but may be wary of downloading what they feel are potential viruses or malware on their device.

In terms of this first assignment, you will be asked to write a few simple programs in the Processing language. These could range from producing basic geometric shapes onscreen, to creating animated gifs or short videos.

Assignment Activity 2: Understand, be able to explain, and reproduce a range of fundamental computer graphics algorithms, including line drawing, triangle drawing, clipping, and curve drawing

Computer graphics algorithms are the fundamental instructions that allow computers to generate digital images. By understanding and being able to reproduce these, you will be taking a big step towards comprehending how 3D graphics software works internally. Algorithms covered in this section will include line drawing, triangle drawing, clipping, and curve drawing.

Line Drawing: The simplest type of graphic is a line. The algorithm for drawing a line is relatively straightforward – the computer simply needs to calculate the points at which the line should be drawn (x1, y1) and (x2, y2). It then draws a pixel between those two points.

Triangle Drawing: A triangle can be defined by its three vertices, (x1, y1), (x2, y2), and (x3, y3). For these three points to make up a triangle, the x coordinate of (x1, y1) must be less than or equal to the x coordinate of both (x2,y2) and (x3, y3). If this was not the case, then the line joining (x1,y1) and (x2,y2) would be parallel to the line joining (x1, y1) and (x3, y3), which means that they are not part of the same triangle.

Triangle Clipping: Once a triangle has been drawn, it is often necessary to clip it so that only the relevant part of the triangle is shown. The algorithm for clipping a triangle is as follows:

• Check if the point (x, y) is within the boundaries of the triangle. If it is not, then discard the triangle.
• Calculate the two coordinates for the edges on each side of the point, (x1, y1) and (x2, y2).
• Calculate the midpoint of these two points. This is done by dividing both x coordinates in half and then dividing both y coordinates in half.
• The final step is to check if the point (x, y) is within the boundaries of the midpoint. If it is, then show the triangle. If it is not, then discard the triangle.

Curve Drawing: Curves can be drawn by using a combination of line segments and arcs. The algorithm for drawing a curve is as follows:

• Define a curve by an equation of the form f(x) = y. This is the equation for a line with x and y as variables.
• Calculate the grid size for this curve. The more grids you have, the better defined the curve will be but it will also take longer to process.
• For each pixel on the curve, calculate the corresponding point on the equation using x and y.
• Draw a line from the start point to the endpoint of each segment.
• Draw an arc from the start point to the endpoint of each segment.

Assignment Activity 3: Understand and use the vector and matrix representations in homogenous coordinate systems, to perform geometric transformations

In this assignment, you will be using two different types of data structures to represent points and shapes in a 3D space. These are a vector and a matrix. This idea is fundamental to computer graphics so it is important to understand how these work.

A vector represents a point in space by its components: x, y, and z. A vector can also be thought of as an arrow, with its length representing the Euclidean distance from the origin and its direction i.e. (1, 0, 0) would point up; (-1, 0, 0) would point to the right; (0, 1, 0) would point towards you, etc.

When two vectors are multiplied together, the result is a new vector that is the sum of the two vectors. This can be represented by the equation:

A × B = (XA + YB, XA – YB, ZA)

This is known as a vector product and it is important to note that it is not commutative i.e. A × B is not the same as B × A.

A matrix represents a shape in space by its dimensions and the coordinates of its vertices. It can be thought of as a grid, with each row and column representing a point in space. The following image shows a 2×2 matrix that represents a square:

The following are the basic operations that can be performed on a matrix:

Matrix Multiplication: A × B = C

This is the same as vector multiplication i.e. the result is a new vector that is the sum of the two vectors.

Matrix Inverse: A-1 = B

This is the same as the back-substitution method for solving a system of linear equations.

A matrix can be multiplied with another matrix using this equation:

C = A × B

The order in which two matrices are multiplied is important i.e. it does not commute! The following examples show the different results when multiplying a matrix with another matrix in different orders.

It should be noted that order does not matter when multiplying a matrix by a scalar but it does for matrices of the same size. For example, if you have two matrices A and B which both have dimensions 2 x 3, then C = A × 3B will produce the same result as C = 3B × A.

Now that you understand the basic concepts of vectors and matrices, let’s look at how they can be used to perform geometric transformations. In order to do this, we need to first define what a transformation is. A transformation can be thought of as a function that takes in a point as an input and produces a new point as an output. The original point is known as a vertex and the new point is known as a transformed vertex. For example, if you take a given square and rotate it clockwise by 45 degrees, the rotated square is the result of this transformation:

In computer graphics, transformations can be represented mathematically by 4×4 matrices. These matrices contain the coordinates of the vertices in the original shape and the transformed shape. The following image shows the matrix that represents the rotation transformation above:

As you can see, the matrix contains the coordinates of all four vertices in both the original and rotated shapes. In addition, it also contains a value of 1, 2 or -1 in the diagonal for clockwise rotations.

This is known as the Homogenous Coordinate System and it uses homogeneous coordinates to express geometric transformations. A point of this space is represented by four parameters (x, y, z, w), which together make up a 4D vector. The fourth component (w) is used to specify the type of transformation. If w is equal to 0, then the point is considered to be in the original space. If w is equal to 1, 2, or -1, then the point is in the transformed space and has undergone a rotation, scaling, or shearing transformation, respectively.

This may all seem a bit confusing, but it will make more sense if you see some examples of transformations in action. The following images show the same 4×4 matrix representing different geometric transformations:

To summarize this discussion on geometric transformations; A point can be represented by its coordinates (x, y, z), and these points are connected to form lines which in turn, form shapes. A transformation is a function that takes in a vertex as input and produces a new vertex as an output. The coordinates of the vertices in the original shape are stored in a 4×4 matrix and the transformed shape is also stored in this matrix.

Assignment Activity 4: Understand and explain the human visual system, its limitations, and the implications these limitations have on representations of color, display resolution, and quantization. Describe a number of color spaces and their relative merits. Explain the basics of the key display technologies in current use

The human visual system is a complex and fascinating thing. It plays a crucial role in our ability to interact with the world around us, and it has some unique limitations which need to be taken into account when designing representations of color, display resolution, and quantization. In this section, we will take a closer look at what these limitations are and explore some of the ways in which they can be overcome.

The human visual system is limited in its ability to detect small differences in color. This means that when we are looking at an image, we may not be able to distinguish between two shades of blue that are only a few pixels apart. This limitation is known as the Nyquist-Shannon Sampling Theorem and it tells us that a pixel must be at least 1/2 a wavelength away from its neighbors in order to produce a detectable change. In this case, the blue pixels shown above will not be differentiated by the human visual system which is why some anti-aliasing techniques are employed to smooth out jagged edges and make continuous-tone images more realistic:

The human visual system is also limited in its ability to precisely distinguish between two objects that are very close together. This means that we may not be able to tell the difference between two shades which are next to each other and this is known as the spatial or contrast sensitivity function. In order for us to be able to differentiate between two pixels that are less than a wavelength apart, they must be spaced at least one-third of a wavelength away from their neighbors. For example:

We can clearly see that the human visual system is not going to be able to distinguish between the blue and green pixels without some form of anti-aliasing. In addition, the contrast between these two colors will also be reduced if they are placed too close together.

The human visual system is also limited in its resolution or ability to see fine details. This means that we can only see a certain number of pixels at a time and any detail smaller than this will not be visible. The maximum number of pixels that the human visual system can see is called the Nyquist limit and it is about 1,500 per inch. This limitation affects both the spatial resolution and the temporal resolution of our vision. The spatial resolution is the maximum number of pixels that we can see in a single image and the temporal resolution is the maximum number of pixels that we can see over time. The display technology that is currently used has a limited spatial resolution and this means that the images must be enlarged in order to make them more visible.

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