computer graphics midterm

coordinate system

is defined by an origin point and the orientation and scale of a set of coordinate axes.

point 𝒑 = (𝒙, 𝒚)

refers to a location in space, specified relative to a coordinate system.

vector 𝒗 = 〈𝒎, 𝒏〉

refers to a displacement—an amount of change in each coordinate—and is typically drawn as an arrow pointing along the direction of displacement

Initial point or tail

– the point where the arrow begins

Terminal point or head

– the point where the arrow ends; indicates the result when the displacement has been applied to the initial point.

Length or magnitude

– the distance between the initial and terminal points of the vector; can be calculated from the components of the vector.

standard position

A vector whose initial point is located at the origin (when a coordinate system is specified) is said to be in

scalars

Individual numbers (that are not part of a point or vector) are often called

Pythagorean theorem.

you can determine the length/magnitude of a vector using the

matrix

is a rectangular array of values called elements that are typically accessed by means of subscripts

first subscript

refers to the row number,

second one

refers to the column number, with the subscripts starting at 0.

 identity matrix

contains all zeros, with ones along the diagonal. Any point or matrix multiplied by the identity matrix is unchanged.

transpose

of a matrix is computed by interchanging its rows and columns

scalar multiplication

you can multiply a scalar by each element of the matrix using the dot operator.

matrix multiplication

you can only multiply matrices if the number of columns on the left-hand side matrix is equal to the number of rows on the righthand side matrix. To multiply matrices, multiply the elements of each row of the first matrix by the elements of each column in the second matrix. Then, add the products.

A geometric transformation

refers to a change applied to an object in terms of size, orientation, or position

Scaling:

The object’s dimensions are either expanded or compressed to change its size. A scale matrix can also be used to move points towards or away from the origin. By scaling relative to the origin, all coordinates of the points defining an entity are multiplied by the same factor.

Translation

: The object is moved to another position or location on the screen. All coordinates of the points defining an entity are modified by adding the same vector quantity.

Rotation:

The object is moved around a fixed point at a given angle. In 2D, a rotation transformation rotates vectors around the origin point. In 3D, rotations are performed around a line rather than a point.

scaling factor

is used to determine whether the size of an object will be increased or reduced. A—— greater than 1 implies that the object’s size will be increased. If the scaling factor is less than 1, the object size will be reduced.

Rotation

In 3D, there are three (3) types of rotation: x-axis rotation, y-axis rotation, and z-axis rotation. Given the initial x, y, z coordinates of an object and the rotation angle, you can determine the new coordinates by creating 4x4 matrices with combinations of the sine and cosine functions.

frustum or truncated pyramid

perspective projection, the shape of the viewable region is

near and far distance

refer to distances from the viewer (along the z-axis). These two set absolute bounds on what could potentially be seen by the viewer. Any points beyond this range will not be rendered. However, not everything between these bounds will be visible.

angle of view

measures how much of the scene is visible to the viewer. It is defined as the angle between the top and bottom planes of the frustum if those planes were extended to the origin

aspect ratio

is used to specify the shape of the frustum which depicts the frustum as viewed from the front (along the negative z-axis).

projection window:

a flat rectangular region in space corresponding to the rendered image that will be displayed on the computer screen. The p—-corresponds to the smaller rectangular side of the frustum, the side nearest to the origin.