Download Transforms - Lecture`s of computer graphics

Trigonometry, Coordinate
system and Vectors
Computer Graphics
Trigonometry
• Tri-gon-metry: measurement of three sided polygons i.e. triangles
• Angular measurement: degree and radians
• The degree (or sexagesimal) unit of measure derives from defining
one complete rotation as 360◦
• Each degree divides into 60 minutes, and each minute divides into 60
seconds.
• The radian of angular measure does not depend upon any arbitrary
constant—it is the angle created by a circular arc whose length is
equal to the circle’s radius.
Trigonometry
• The perimeter of a circle is 2πr, 2π radians correspond to one
complete rotation. As 360◦ correspond to 2π radians, 1 radian
equals 180◦ /π which is approximately 57.3◦
Trigonometry
Trigonometric Ratios (For unit Circle)
Trigonometric Identities
Coordinate System
• Coordinate Reference Frames
• Cartesian coordinate system
• x, y, z
• Non-Cartesian coordinate system
• Polar, Spherical, Cylindrical
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2D Cartesian Plan
• The Cartesian plane provides a mechanism for visualizing pairs of
related variables graphically. The variables are normally x and y, as
used in functions such as
y = 0.5x + 1.
• The set of points create a familiar
straight line associated with
equations of the form
y = mx + c.
2D Cartesian Plan
• Any point P on the Cartesian plane is identified by an ordered pair of
numbers (x, y) where x and y are called the Cartesian coordinates of P.
• X being independent variable and Y, the dependent.
• a, b, c are used to represent the coefficients
y = ax2 +bx +c
The point where Cartesian coordinates intersect, makes an angle of 90
degree, known as Origin.
Function Graph
• Linear functions are straight lines
• quadratics are parabolas
• cubics have an “S” shape
• trigonometric functions often possess a wave-like trace.
• Such graphs are used in animation to control the movement of
objects, lights and virtual camera
A function curve relating brightness to frame
number
A function curve relating brightness to frame
number
an example where the horizontal axis marks the progress of time in
animation frames, and the vertical axis records the corresponding
brightness of a virtual light source. Such a function forms part of the
animator’s user interface, and communicates in a very intuitive manner
the brightness of the light source for every frame of animation. The
animator can then make changes to the function with the aid of
interactive software tools.
Geometric Shapes
• In computer graphics, 2D shapes and 3D objects have a numerical
description of some sort. Shapes include polygons, circles, arbitrary
curves, mathematical functions, fractals, etc., and objects can be
faceted, smooth, bumpy, furry, gaseous, etc.
Polygonal Shapes
• A polygon is constructed from a chain of vertices (points). A straight
line is assumed to connect each pair of neighboring 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.
Arithmetical operations can be performed on
coordinate points of polygon.
Double the values of x
and y and redraw
polygon: shape will be
preserved but size of the
polygon will double.
By dividing x and y by 2
will generate a smaller
polygon with same
shape.
Area of polygonal shape
• The area of a polygonal shape is readily calculated from its chain of
coordinates. For example, given the following list of coordinates:
Distance Formula
Distance between two points
The theorem of Pythagoras (distance formula) is used to calculate the
distance between two points. To find the distance between two
arbitrary points P1(x1, y1) and P2(x2, y2).
The distance Δx = x2 −x1 and Δy = y2 −y1. Therefore, the distance d
between P1 and P2 is given by
3D Cartesian Plan
• To locate a point in space, 3D coordinate system is used
Right Handed axial system
Distance Formula for 3D
• The theorem of Pythagoras in 3D is a natural extension of the 2D rule.
In fact, it even works in higher dimensions.
• Given two arbitrary points P1(x1, y1, z1) and P2(x2, y2, z2), we
compute Δx = x2 − x1,
• Δy = y2 − y1
• Δz = z2 − z1,
• from which the distance d between P1 and P2 is given by
Euler’s Rule
In 1619, Descartes discovered quite a nice relationship between
vertices, edges and the faces of a 3D polygonal object
Faces + vertices = edges+ 2.
Example:
Cube: 12 edges
6 Faces
8 vertices
Coordinate System
• Coordinate Reference Frames
Cartesian coordinate system
 x, y, z
Non-Cartesian coordinate system
 Polar, Spherical, Cylindrical
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Non-Cartesian Coordinate System
• Polar Coordinates
Polar coordinates are used for handling data containing angles, rather
than linear offsets.
Non-Cartesian Coordinate System
point P(x, y) has equivalent polar
coordinates P(ρ,θ), where:
The point (3, 4) have polar
coordinates;
Non-Cartesian Coordinate System
• Spherical Polar Coordinates
the point P(x, y, z) has equivalent
polar coordinates P(ρ,φ,θ)
Non-Cartesian Coordinate System
For example, the point
(3, 4, 0) has spherical polar
coordinates
(5, 90◦, 53.13◦)
Non-Cartesian Coordinate System
• Cylindrical coordinates
the point P(x, y, z) has equivalent
cylindrical coordinates P(ρ,θ,z),
Where,
Example
For example, the point (3, 4, 6) has cylindrical
coordinates (5, 53.13◦, 6):
Vectors
Introduction to vectors covers 2D and 3D vectors, unit vectors, position
vectors, Cartesian vectors, vector magnitude, vector products, and area
calculations. It also shows how vectors are used in lighting calculations
and back-face detection.
• Vectors provide us with some powerful techniques for computing
angles between lines and the orientation of surfaces. They also
provide a coherent framework for computing the behaviour of
dynamic objects in computer animation and illumination models in
rendering.
Scalar quantity
• Scalar Quantity: a quantity that can be completely described by their
magnitudes. For example, height, shoe size, length of tower etc.
• In computer graphics, scalar quantities include colour, height, width,
depth, brightness, number of frames, etc.
• Scalar quantities are represented by small alphabets.
Vector quantities
There are some things that require more than one number to represent
them: wind, force, weight, velocity and sound, are just a few examples.
These cannot be represented accurately by a single number.
Example:
Any sailor knows that wind has a magnitude and a direction. The force we
use to lift an object also has a value and a direction.
Velocity of a moving object is measured in terms of its speed (e.g. miles per
hour), and a direction such as northwest. Sound, too, has intensity and a
direction. Such quantities are called vectors. In computer graphics, vectors
generally require two or three numbers, and these are the only type we
consider.
Vector quantities are represented by bold fonts.
2D vectors
• When a scalar variable is assigned a value, we use the standard
algebraic notation:
x=3
• However, a vector can be stored as a column or as a row—and in this
text I use column vectors:
3
n=
5
A row vector places the components horizontally.
n = [3 5]
Graphical Representation of Vectors
As vectors encode direction, as well as magnitude, an arrow is often
used to indicate direction, and a number to represent magnitude. Such
a scheme is often used in weather maps.
Using Cartesian coordinates, the line’s direction is determined by
identifying the vector’s tail and then measuring its components along
the x- and y-axis.
2D Vector with magnitude and Direction
The vector r has its tail defined by (x1, y1) = (1, 2), and its head by
(x2, y2) = (3, 4). Vector s has its tail defined by (x3, y3) = (5, 3), and its
head by (x4, y4) = (3, 1).
• Two vectors r and s
have the same magnitude but
opposite directions
Find the direction and
Magnitude of the vectors r and s
Magnitude of the Vector
The magnitude or length of a vector r is written |r| and computed
using the theorem of Pythagoras:
Quiz # 1
• Calculate the magnitude and direction of the given vector having
coordinates
(3, 5) and (9,6)
Question 2: Find the polar coordinates of (5,7)