Introduction to Computer Graphics: Foundations and Techniques

1. CS 551 / 645: Introductory Computer Graphics David Luebke cs551@cs.virginia.edu http://www.cs.virginia.edu/~cs551 David Luebke 11/15/2014

2. 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 11/15/2014

3. 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 11/15/2014

4. 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 11/15/2014

5. 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 11/15/2014

6. 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 11/15/2014

7. 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 11/15/2014

8. 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 11/15/2014

9. Vector Spaces • Two types of elements: • Scalars (real numbers): a, b, g, d, … • Vectors (n-tuples):u, v, w, … • Supports two operations: • Addition operation u + v, with: • Identity 0v + 0 = v • Inverse -v + (-v) = 0 • Scalar multiplication: • Distributive rule: a(u + v) = a(u) + a(v) (a + b)u = au + bu David Luebke 11/15/2014

10. Vector Spaces • A linear combination of vectors results in a new vector: v= a1v1 + a2v2 + … + anvn • If the only set of scalars such that a1v1 + a2v2 + … + anvn = 0 is a1 = a2 = … = a3 = 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 11/15/2014

11. u+v y v u x 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”: David Luebke 11/15/2014

12. 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 11/15/2014

13. 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 11/15/2014

14. 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 fromPtoQ • Vector-point addition: P + v = Q • Result is a new point • Note that the addition of two points is not defined Q v P David Luebke 11/15/2014

15. 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 11/15/2014

16. Affine Lines: An Aside • Parametric representation of a line with a direction vector d and a point P1 on the line: P(a) = Porigin + ad • Restricting 0aproduces a ray • Setting d to P - Q and restricting 0a 1 produces a line segment between P and Q David Luebke 11/15/2014

17. 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 v θ u David Luebke 11/15/2014

18. 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 David Luebke 11/15/2014

19. 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 11/15/2014

20. Matrices • By convention, matrix elementMrc is located at row r and column c: • By (OpenGL) convention, vectors are columns: David Luebke 11/15/2014

21. Matrices • Matrix-vector multiplication applies a linear transformation to a vector: • Recall how to do matrix multiplication David Luebke 11/15/2014

22. 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 matrixI has no effect in multiplication • Some (not all) matrices have an inverse: David Luebke 11/15/2014

23. David Luebke 11/15/2014

24. The End • Next: drawing lines on a raster display • Reading: FvD 3.2 David Luebke 11/15/2014