Topics Covered:
The Cartesian XY-plane, Function Graphs, Geometric Shapes, Polygonal Shapes, Areas of Shapes, Theorem
of Pythagoras in 2D, Coordinates, Theorem of Pythagoras in 3D, 3D Polygons, Euler’s -------------------------------------------------------------------------------------------------------------------------------------------------------------------
Mathematics for Computer Graphics
Computer graphics contains many areas of specialism, such as data visualization, 2D computer animation,
film special effects, computer games and 3D computer animation.
Fortunately, not everyone working in computer graphics requires a knowledge of mathematics, but those
that do, often look for a book that introduces them to some basic ideas of mathematics, without turning
them into mathematicians. This is the objective of this book. Over the following 18 chapters I introduce the
reader to some useful mathematical topics that will help them understand the software they work with, and
how to solve a wide variety of geometric and algebraic problems.
These topics include numbers systems, algebra, trigonometry, 2D and 3D geometry, vectors, equations,
matrices, determinants and calculus.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------
The Cartesian Plane
The Cartesian plane provides a mechanism for locating points with a unique, ordered pair of numbers (x, y)
as shown in Fig. 5.1,where P has coordinates (3, 2) and Q has coordinates (−4,−2). The point (0, 0) is
called the origin. As previously mentioned,
Descartes suggested that the letters x and y should be used to represent variables, and letters at the other
end of the alphabet should stand for numbers. Which is why equations such as y = ax2 + bx + c,, are written
this way.
The axes are said to be oriented as the x-axis rotates anticlockwise towards the y-axis. They could have
been oriented in the opposite sense, with the y-axis rotating anticlockwise towards the x-axis.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------
Function Graphs
When functions such as
are drawn as graphs, they create familiar shapes that permit the function to be easily identified. Linear
functions are straight lines; quadratics are parabolas; cubics havean ‘S’ shape; and trigonometric functions
often possess a wave-like trace. Figure 5.2 shows examples of each type of function.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
Shape Representation
The Cartesian plane also provides a way to represent 2D shapes numerically, which permits them to be
manipulated mathematically. Let’s begin with 2D polygons and show how their internal area can be
calculated.
5.5.1 2D Polygons
A polygon is formed from a chain of vertices (points) as shown in Fig. 5.3. A straight line is assumed to
connect each pair of neighbouring vertices; intermediate points on the line are not explicitly stored. There
is no convention for starting a chain of vertices, but software will often dictate whether polygons have a
clockwise or anticlockwise vertex sequence.
We can now subject this list of coordinates to a variety of arithmetic and mathematical operations. For
example, if we double the values of x and y and redraw the vertices, we discover that the shape’s geometric
integrity is preserved, but its size is doubled relative to the origin. Similarly, if we divide the values of x and
y by 2, the shape is still preserved, but its size is halved relative to the origin. On the other
hand, if we add 1 to every x-coordinate, and 2 to every y-coordinate, and redraw the vertices, the shape’s
size remains the same but is displaced 1 unit horizontally and 2 units vertically.
The Cartesian XY-plane, Function Graphs, Geometric Shapes, Polygonal Shapes, Areas of Shapes, Theorem
of Pythagoras in 2D, Coordinates, Theorem of Pythagoras in 3D, 3D Polygons, Euler’s -------------------------------------------------------------------------------------------------------------------------------------------------------------------
Mathematics for Computer Graphics
Computer graphics contains many areas of specialism, such as data visualization, 2D computer animation,
film special effects, computer games and 3D computer animation.
Fortunately, not everyone working in computer graphics requires a knowledge of mathematics, but those
that do, often look for a book that introduces them to some basic ideas of mathematics, without turning
them into mathematicians. This is the objective of this book. Over the following 18 chapters I introduce the
reader to some useful mathematical topics that will help them understand the software they work with, and
how to solve a wide variety of geometric and algebraic problems.
These topics include numbers systems, algebra, trigonometry, 2D and 3D geometry, vectors, equations,
matrices, determinants and calculus.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------
The Cartesian Plane
The Cartesian plane provides a mechanism for locating points with a unique, ordered pair of numbers (x, y)
as shown in Fig. 5.1,where P has coordinates (3, 2) and Q has coordinates (−4,−2). The point (0, 0) is
called the origin. As previously mentioned,
Descartes suggested that the letters x and y should be used to represent variables, and letters at the other
end of the alphabet should stand for numbers. Which is why equations such as y = ax2 + bx + c,, are written
this way.
The axes are said to be oriented as the x-axis rotates anticlockwise towards the y-axis. They could have
been oriented in the opposite sense, with the y-axis rotating anticlockwise towards the x-axis.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------Function Graphs
When functions such as
are drawn as graphs, they create familiar shapes that permit the function to be easily identified. Linear
functions are straight lines; quadratics are parabolas; cubics havean ‘S’ shape; and trigonometric functions
often possess a wave-like trace. Figure 5.2 shows examples of each type of function.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------Shape Representation
The Cartesian plane also provides a way to represent 2D shapes numerically, which permits them to be
manipulated mathematically. Let’s begin with 2D polygons and show how their internal area can be
calculated.
5.5.1 2D Polygons
A polygon is formed from a chain of vertices (points) as shown in Fig. 5.3. A straight line is assumed to
connect each pair of neighbouring vertices; intermediate points on the line are not explicitly stored. There
is no convention for starting a chain of vertices, but software will often dictate whether polygons have a
clockwise or anticlockwise vertex sequence.
We can now subject this list of coordinates to a variety of arithmetic and mathematical operations. For
example, if we double the values of x and y and redraw the vertices, we discover that the shape’s geometric
integrity is preserved, but its size is doubled relative to the origin. Similarly, if we divide the values of x and
y by 2, the shape is still preserved, but its size is halved relative to the origin. On the other
hand, if we add 1 to every x-coordinate, and 2 to every y-coordinate, and redraw the vertices, the shape’ssize remains the same but is displaced 1 unit horizontally and 2 units vertically.
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