This chapter defines system software and discusses two types of system software: operating systems and utility programs. You learn what an operating system is and explore user interfaces, operating systems features, and operating system functions. | Computer Graphics Lecture 11 Fasih ur Rehman Last Class Geometric Objects Vector Space Affine Space Today’s Agenda Geometric Objects Vector Space Affine Space Basic Geometries Notation Greek letters α, β, γ , . . . denote scalars; uppercase letters P, Q, R, . . . denote points; lowercase letters u, v, w, . . . denote vectors. Point Specifies a location in 3D space and is represented by three coordinates as (x, y, z). In 2D space, however, only two coordinates will be needed The point is infinitely small and Does not possess any shape (x,y,z) Vector Represented by 3 coordinates and Specifies Magnitude () Direction Has no location (dx, dy, dz) Vector Space Vectors define a vector space They support vector addition Commutative and associative Possess identity and inverse They support scalar multiplication Associative, distributive Possess identity Affine Spaces Vector spaces lack position and distance They have magnitude and direction but no location Combine the point and vector . | Computer Graphics Lecture 11 Fasih ur Rehman Last Class Geometric Objects Vector Space Affine Space Today’s Agenda Geometric Objects Vector Space Affine Space Basic Geometries Notation Greek letters α, β, γ , . . . denote scalars; uppercase letters P, Q, R, . . . denote points; lowercase letters u, v, w, . . . denote vectors. Point Specifies a location in 3D space and is represented by three coordinates as (x, y, z). In 2D space, however, only two coordinates will be needed The point is infinitely small and Does not possess any shape (x,y,z) Vector Represented by 3 coordinates and Specifies Magnitude () Direction Has no location (dx, dy, dz) Vector Space Vectors define a vector space They support vector addition Commutative and associative Possess identity and inverse They support scalar multiplication Associative, distributive Possess identity Affine Spaces Vector spaces lack position and distance They have magnitude and direction but no location Combine the point and vector primitives Permits describing vectors relative to a common location A point and three vectors define a 3-D coordinate system Point-point subtraction yields a vector Point – Vector Operations Point – point subtraction yields v = P – Q Point – Vector addition yields P = Q + v Vector Addition We can also use this visualization to show that for any three points P, Q, and R Lines Consider all points of the form P(α) = P0 + α d where P0 is an arbitrary point d is an arbitrary vector and α is a scalar (that can vary over a range) This relation can be interpreted as the set of all points that pass through P0 in the direction of the vector d Line (Slop – Intercept Form) Euclidean Affine Spaces Allows to compute distance and angles Dot product: The dot product of two vectors is a scalar. Let v1 and v2 be two vectors Dot Products Used to compute length (magnitude) of the vector Dot Products Normalization (finding unit vector) Dot Products Computing angle between two vector Dot Products Checking for .
