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# Basic theory of curve and surface - PowerPoint PPT Presentation

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 .

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### Basic theory of curve and surface

Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

• Parametric

• Non-parametric

• Explicit

• Implicit

x = x(u), y = y(u)

• y = f(x)

• f(x, y) = 0

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• Example - circle

• Parametric

• Non-parametric

• Explicit

• Implicit

x = R cos, y = R sin

• y = R2 – x2

• x2 + y2 – R2 = 0

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

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• y = f(x)

• Only one y value for each x value

• Cannot represent closed or multiple-valued curves such as circle

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f(x,y) = 0

ax2 + bxy + cy2 + dx + ey + f = 0

• Advantages – can produce several type of curve – set the coefficients

• Not sure which variable to choose as the independent variable

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• 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

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• Express relationship for the x, y and z coordinates not in term of each other but of one or more independent variable (parameter).

• 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

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• 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.

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• easy to express in the form of vectors and matrices

• Inherently bounded.

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t2

t4

t5

t6

t1

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

x

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b

a

b

a

ray

b

a

Line segment

Parametric curve

• Fundamental geometric objects – lines, rays and line segment

All share the same parametric representation

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a

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  

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• 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

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• 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

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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

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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

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• 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.

hyperbola

parabola

ellipse

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• 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

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Parametric curve (circle)

• If t = 0.125  a 1/8 circle

• t = 0.25  a 1/4 circle

• t = 0.5  a ½ circle

t = 1  a circle

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• Circle with center at (xc, yc)

• x(t) = R cos(2t) + xc,

• y(t) = R sin(2t) + yc ,

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• Ellipse

• 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

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• Shape (based upon parametric equation)

• Location (based upon center point)

• Size

• Arc (based upon parameter range)

• Radius (a coefficient to unit value)

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• 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

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• Tangent vector

• Vector tangent to the slope of curve at a given point

• Tangent line

• The line that contains the tangent vector

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• 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|

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• 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

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• 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).

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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))

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• 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)))

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• Check / prove

• Let say, t = 0,

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

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• 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)

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• 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 = ?

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P

P

• 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

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• 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 ?

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• 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.

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• 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.

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• Typically represented

• A series of simpler curve (each defined by a single equation) pieced together at their endpoints.(piecewise construction)

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• Typically represented

• Simple curve may be linear or polynomial

• Equation for simpler curves based on control points (data points used to define the curve).

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a) Desired curve

b) User places points

c) The algorithm generates many points along a “nearby” curve

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• 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.