人机接口与图形学双语Mathematicalrinciles.ppt

* Mathematical  Foundation Department of CS E, East China University of Science Technology Mathematical Foundations Appendix partly I’ll give a brief, informal review of some of the mathematical tools we’ll employ Geometry (2D, 3D) Trigonometry Vector spaces Points, vectors, and coordinates Dot and cross products Basis Vectors and Matrix 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 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) Slope-intercept Line Equation Slope =m = rise / run Slope = (y - y1) / (x - x1)  = (y2 - y1) / (x2 - x1) Solve for y: y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1 or: y = mx + b x y P2 = (x2, y2) P1 = (x1, y1) P = (x, y) Parametric Line Equation Given points P1 = (x1, y1) and P2 = (x2, y2) x = x1 + t(x2 - x1) y = y1 + t(y2 - y1) When: t=0, we get (x1, y1) t=1, we get (x2, y2) (0t1), we get pointson the segment between(x1, y1) and (x2, y2) x y P2 = (x2, y2) P1 = (x1, y1) Other helpful formulas Length = sqrt {(x2 - x1)2 + (y2 - y1)2} Midpoint, p2, between p1 and p3 p2 = ((x1 + x3) / 2, (y1 + y3) / 2)) Two lines are perpendicular if: M1 = -1/M2 cosine of the angle between them is 0 Scalar Spaces Scalars: a, b, … Addition and multiplication (+ and h) operations defined Scalar operations are Associative: a + (b + g) = (a + b) + g Commutative: a + b = b + a a h b = b h a Distributive: a h(b h g) = (a h b) h g  a h(b + g) = (a h b) + (a h g) Scalar Spaces Additive Identity = 0 a + 0 = 0 + a = a Multiplicative Identity = 1 a h 1 = 1 h a = a Additive Inverse = -a a + (-a) = 0 Multiplicative Inverse= a-1 a h a-1 = 1 Vector Spaces Two types of elements: Scalars