Basic theory of curve and surface

Geometric representation Parametric Non-parametric Explicit Implicit x = x(u), y = y(u) y = f(x) f(x, y) = 0

Geometric representation Example - circle Parametric Non-parametric Explicit Implicit x = R cos, y = R sin y = R2 – x2 x2 + y2 – R2 = 0

Geometric representation Each form has its own advantages and disadvantages, depending on the application for which the equation is used.

Non-parametric (explicit) y = f(x) Only one y value for each x value Cannot represent closed or multiple-valued curves such as circle

Non-parametric (implicit) f(x,y) = 0 ax2 + bxy + cy2 + dx + ey + f = 0 Advantages – can produce several type of curve – set the coefficients Disadvantages Not sure which variable to choose as the independent variable

Non-parametric (cont) Disadvantages Non-parametric elements are axis dependant, so the choice of coordinate system affects the ease of using the element and calculating their properties. Problem  if the curve has a vertical slope (infinity). They represent unbounded geometry e.g ax + by + c = 0 define an infinite line

parametric Express relationship for the x, y and z coordinates not in term of each other but of one or more independent variable (parameter). Advantages Offer more degrees of freedom for controlling the shape (non-parametric) y= ax3 + bx2 + cx + d (parametric) x = au3 + bu2 + cu + d y = eu3 + fu2 + gu + h

Parametric (cont) Advantages (cont) Transformations can be performed directly on parametric equations. Parametric forms readily handle infinite slopes without breaking down computationally dy/dx = (dy/du)/ (dx/du) Completely separate the roles of the dependent and independent variable.

Parametric (cont) Advantages (cont) easy to express in the form of vectors and matrices Inherently bounded. Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

Parametric curve Use parameter to relate coordinate x and y (2D). Analogy Parameter t (time) – [ x(t), y(t) as the position of the particle at time t ] y t3 t2 t4 t5 t6 t1 x

Parametric curve Fundamental geometric objects – lines, rays and line segment a b line a b ray a b Line segment All share the same parametric representation

Parametric line a = (ax, ay), b = (bx, by) x(t ) = ax + (bx - ax)t y(t) = ay + (by - ay)t Parameter t is varied from 0 to 1 to define all point along the line When t = 0, the point is at “a”, as t increases toward 1, the point moves in a straight line to b. For line segment : 0  t  1 For line : -  t   For ray : 0  t   a b

Parametric line Example A line from point (2, 3) to point (-1, 5) can be represented in parametric form as x(t) = 2 + (-1 – 2)t = 2 – 3t y(t) = 3 + (5 – 2)t = 3 + 3t

Parametric line Positions along the line are based upon the parameter value E.g midpoint of a line occurs at t = 0.5 Exercise : Find the parametric form for the segment with endpoints (2, 4, 1) and (7, 5, 5). Find the midpoint of the segment by using t = 0.5

Parametric line Answer: Parametric form: x(t) = 2 + (7 –2)t = 2 + 5t y(t) = 4 + (5 – 4)t = 4 + t z(t) = 1 + (5 – 1)t = 1 + 4t

Parametric line Answer Midpoint x(0.5) = 2 + 5(0.5) = 5.5  6 Y(0.5) = 4 + (0.5) = 4.5  5 Z(0.5) = 1 + 4(0.5) = 3  3

Parametric curve (conic section) Another basic example Conic section - the curves / portions of the curves, obtained by cutting a cone with a plane. The section curve may be a circle, ellipse, parabola or hyperbola. ellipse hyperbola parabola

Parametric curve (circle) The simplest non-linear curve - circle - circle with radius R centered at the origin x(t) = R cos(2t) y(t) = R sin(2t) 0  t  1

Parametric curve (circle) If t = 0.125  a 1/8 circle Circular arc t = 0.25  a 1/4 circle t = 0.5  a ½ circle t = 1  a circle

Parametric curve (circle) Circle with center at (xc, yc) x(t) = R cos(2t) + xc, y(t) = R sin(2t) + yc ,

Parametric curve Ellipse Hyperbola parabola b x(t) = a cos(2t) y(t) = b sin(2t) Hyperbola x(t) = a sec(t) y(t) = b tan(t) parabola x(t) = at2 y(t) = 2at b a b a

Control for this curve Shape (based upon parametric equation) Location (based upon center point) Size Arc (based upon parameter range) Radius (a coefficient to unit value) Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

Parametric curve Generally A parametric curve in 3D space has the following form F: [0, 1] (x(t), y(t), z(t)) where x(), y() and z() are three real-valued functions. Thus, F(t) maps a real value t in the closed interval [0,1] to a point in space for simplicity, we restrict the domain to [0,1]. Thus, for each t in [0,1], there corresponds to a point (x(t), y(t), z(t) ) in space. If z( ) is removed - ? A curve in a coordinate plane

Tangent vector and tangent line Vector tangent to the slope of curve at a given point Tangent line The line that contains the tangent vector

Compute tangent vector F(t) = (x(t), y(t), z(t)) Tangent vector : F’(t) = (x’(t), y’(t), z’(t)) Where x’(t)= dx/dt, y’(t)= dy/dt, z’(t)= dz/dt Magnitude /length If vector V = (a, b, c)  |V| =  a2 + b2 + c2 Unit vector Uv = V / |V|

Compute tangent line Tangent line at t is either F(t) + uF’(t) or F(t) + u(F’(t)/|F’(t)|)  if prefer unit vector u is a parameter for line

example Question: - given a Circle, F(t) = (Rcos(2t), R sin(2t)) , 0  t  1 Find tangent vector at t and tangent line at F(t).

example Answer dx = Rcos(2t), dy = R sin(2t) x’(t) = dx/dt = - 2 Rsin (2t), y’(t) = dy/dt = 2Rcos(2t) Tangent vector = (- 2 Rsin (2t), 2Rcos(2t))

example Answer Tangent line F(t) + u(F’(t)) (Rcos(2t), R sin(2t)) + u(- 2 Rsin (2t), 2Rcos(2t)) (Rcos(2t) + u(- 2 Rsin(2t))) , (R sin(2t) + u(2Rcos(2t)))

Example Check / prove Let say, t = 0, R Tangent vector = (- 2 Rsin (2t), 2Rcos(2t)) = (0, 2R) tangent line = (Rcos(2t) + u(- 2 Rsin(2t))) , (R sin(2t) + u(2Rcos(2t))) = (R, u(2R)) R

Tangent vector Slope of the curve at any point can be obtained from tangent vector. Tangent vector at t = (x’(t), y’(t)) Slope at t = dy/dx = y’(t)/x’(t)

curvature The curvature at a point measures the rate of curving (bending) as the point moves along the curve with unit speed When orientation is changed the curvature changes its sign, the curvature vector remains the same Straight line  curvature = ?

curvature Circle is tangent to the curve at P lies toward the concave or inner side of the curve at P Curvature = 1/r , r radius

curvature The curvature at u, k(u), can be computed as follows: k(u) = | f'(u) × f''(u) | / | f'(u) |3 How about curvature of a circle ?

Curve use in design Engineering design requires ability to express complex curve shapes (beyond conic) and interactive. Bounding curves for turbine blades, ship hulls, etc Curve of intersection between surfaces.

Curve use in design A design is “GOOD” if it meets its design specifications : These may be either : Functional - does it works. Technical - is it efficient, does it meet certain benchmark or standard. Aesthetic - does it look right, this is both subjective and opinion is likely to change in time or combination of both.

Representing complex curves Typically represented A series of simpler curve (each defined by a single equation) pieced together at their endpoints.(piecewise construction)

Representing complex curves Typically represented Simple curve may be linear or polynomial Equation for simpler curves based on control points (data points used to define the curve).

An interactive curve design a) Desired curve b) User places points c) The algorithm generates many points along a “nearby” curve

An interactive curve design Interactive design consists of the following steps Lay down the initial control points Use the algorithm to generate the curve If the curve satisfactory, stop. Adjust some control points Go to step 2.