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CS 551 / 645: Introductory Computer Graphics. David Luebke [email protected] http://www.cs.virginia.edu/~cs551. Administrivia. Office hours: Luebke: 2:30 - 3:30 Monday, Wednesday Olsson 219 Dale / Jinze / Derek: 3 - 4 Tuesday 2 - 3 Thursday 2 - 3 Friday Small Hall Unixlab for now - PowerPoint PPT Presentation
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David Luebke 04/19/23
CS 551 / 645: Introductory Computer Graphics
David Luebke
http://www.cs.virginia.edu/~cs551
David Luebke 04/19/23
Administrivia
Office hours:– Luebke:
2:30 - 3:30 Monday, Wednesday Olsson 219
– Dale / Jinze / Derek: 3 - 4 Tuesday 2 - 3 Thursday 2 - 3 Friday Small Hall Unixlab for now Might move to Olsson 227 later
Assignment 1 postponed till Wednesday
David Luebke 04/19/23
UNIX Section: 4 PM, Olsson 228
Optional UNIX section led by Dale today– Getting around– Using make and makefiles– Using gdb (if time)
We will use 2 libraries: OpenGL and Xforms– OpenGL native on SGIs; on other platforms Mesa– Xforms: available on all platforms of interest
David Luebke 04/19/23
XForms Intro
Xforms: a toolkit for easily building Graphical User Interfaces, or GUIs– See http://bragg.phys.uwm.edu/xforms– Lots of widgets: buttons, sliders, menus, etc.– Plus, an OpenGL canvas widget that gives us a
viewport or context to draw into with GL or Mesa.
Quick tour now You’ll learn the details yourself in
Assignment 1
David Luebke 04/19/23
Mathematical Foundations
FvD appendix gives good review I’ll give a brief, informal review of some of the
mathematical tools we’ll employ– Geometry (2D, 3D)– Trigonometry– Vector and affine spaces
Points, vectors, and coordinates
– Dot and cross products– Linear transforms and matrices
David Luebke 04/19/23
2D Geometry
Know your high-school geometry:– Total angle around a circle is 360° or 2π radians– When two lines cross:
Opposite angles are equivalent Angles along line sum to 180°
– Similar triangles: All corresponding angles are equivalent Corresponding pairs of sides have the same length ratio
and are separated by equivalent angles Any corresponding pairs of sides have same length ratio
David Luebke 04/19/23
Trigonometry
Sine: “opposite over hypotenuse” Cosine: “adjacent over hypotenuse” Tangent: “opposite over adjacent” Unit circle definitions:
– sin () = x– cos () = y– tan () = x/y– Etc…
(x, y)
David Luebke 04/19/23
3D Geometry
To model, animate, and render 3D scenes, we must specify:– Location– Displacement from arbitrary locations– Orientation
We’ll look at two types of spaces:– Vector spaces– Affine spaces
We will often be sloppy about the distinction
David Luebke 04/19/23
Vector Spaces
Two types of elements:– Scalars (real numbers): … – Vectors (n-tuples): u, v, w, …
Supports two operations:– Addition operation u + v, with:
Identity 0 v + 0 = v Inverse - v + (-v) = 0
– Scalar multiplication: Distributive rule:(u + v) = (u) + (v)
( + )u = u + u
David Luebke 04/19/23
Vector Spaces
A linear combination of vectors results in a new vector:
v = 1v1 + 2v2 + … + nvn
If the only set of scalars such that
1v1 + 2v2 + … + nvn = 0
is 1 = 2 = … = 3 = 0
then we say the vectors are linearly independent The dimension of a space is the greatest number of linearly
independent vectors possible in a vector set For a vector space of dimension n, any set of n linearly
independent vectors form a basis
David Luebke 04/19/23
Vector Spaces: A Familiar Example Our common notion of vectors in a 2D plane
is (you guessed it) a vector space:– Vectors are “arrows” rooted at the origin– Scalar multiplication “streches” the arrow, changing
its length (magnitude) but not its direction– Addition uses the “trapezoid rule”:
u+vy
xu
v
David Luebke 04/19/23
Vector Spaces: Basis Vectors
Given a basis for a vector space:– Each vector in the space is a unique linear
combination of the basis vectors– The coordinates of a vector are the scalars from
this linear combination– Best-known example: Cartesian coordinates
Draw example on the board
– Note that a given vector v will have different coordinates for different bases
David Luebke 04/19/23
Vectors And Point
We commonly use vectors to represent:– Points in space (i.e., location)– Displacements from point to point– Direction (i.e., orientation)
But we want points and directions to behave differently– Ex: To translate something means to move it
without changing its orientation– Translation of a point = different point– Translation of a direction = same direction
David Luebke 04/19/23
Affine Spaces
To be more rigorous, we need an explicit notion of position
Affine spaces add a third element to vector spaces: points (P, Q, R, …)
Points support these operations– Point-point subtraction: Q - P = v
Result is a vector pointing from P to Q
– Vector-point addition: P + v = Q Result is a new point
– Note that the addition of two points is not defined
P
Q
v
David Luebke 04/19/23
Affine Spaces
Points, like vectors, can be expressed in coordinates– The definition uses an affine combination– Net effect is same: expressing a point in terms of a
basis
Thus the common practice of representing points as vectors with coordinates (see FvD)
Analogous to equating points and integers in C– Be careful to avoid nonsensical operations
David Luebke 04/19/23
Affine Lines: An Aside
Parametric representation of a line with a direction vector d and a point P1 on the line:
P() = Porigin + d
Restricting 0 produces a ray Setting d to P - Q and restricting 0 1
produces a line segment between P and Q
David Luebke 04/19/23
Dot Product
The dot product or, more generally, inner product of two vectors is a scalar:
v1 • v2 = x1x2 + y1y2 + z1z2 (in 3D)
Useful for many purposes– Computing the length of a vector: length(v) = sqrt(v • v)
– Normalizing a vector, making it unit-length– Computing the angle between two vectors:
u • v = |u| |v| cos(θ)
– Checking two vectors for orthogonality– Projecting one vector onto another
θu
v
David Luebke 04/19/23
Cross Product
The cross product or vector product of two vectors is a vector:
The cross product of two vectors is orthogonal to both
Right-hand rule dictates direction of cross product
1221
1221
1221
21 )(
y x y x
z x z x
z y z y
vv
David Luebke 04/19/23
Linear Transformations
A linear transformation: – Maps one vector to another– Preserves linear combinations
Thus behavior of linear transformation is completely determined by what it does to a basis
Turns out any linear transform can be represented by a matrix
David Luebke 04/19/23
Matrices
By convention, matrix element Mrc is located at row r and column c:
By (OpenGL) convention, vectors are columns:
mnm2m1
2n2221
1n1211
MMM
MMM
MMM
M
3
2
v
v
v
v
1
David Luebke 04/19/23
Matrices
Matrix-vector multiplication applies a linear transformation to a vector:
Recall how to do matrix multiplication
z
y
x
v
v
v
MMM
MMM
MMM
vM
333231
232221
131211
David Luebke 04/19/23
Matrix Transformations
A sequence or composition of linear transformations corresponds to the product of the corresponding matrices– Note: the matrices to the right affect vector first– Note: order of matrices matters!
The identity matrix I has no effect in multiplication
Some (not all) matrices have an inverse: vvMM 1
David Luebke 04/19/23
David Luebke 04/19/23
The End
Next: drawing lines on a raster display Reading: FvD 3.2