basic curve surface

Disediakan oleh Suriati bte Sadimon, GMM, FSKSM, UTM

Basic theory of curve and surface

Geometric representation

• Parametric

• Non-parametric– Explicit

– Implicit

y = f(x)

f(x, y) = 0

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

• Example - circle

• Parametric

• Non-parametric– Explicit

– Implicit

y = R2 – x2

x2 + y2 – R2 = 0

x = R cos, y = R sin

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

Non-parametric (explicit)

• Only one y value for each x value

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

y = f(x)

Non-parametric (implicit)

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

• Disadvantages– Not sure which variable to choose as the

independent variable

f(x,y) = 0ax2 + bxy + cy2 + dx + ey + f = 0

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.

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 ]

Parametric curve

• Fundamental geometric objects – lines, rays and line segment

Line segment

All share the same parametric representation

Parametric linea = (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

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 – 3ty(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

• 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 line

• 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.

Parametric curve (conic section)

ellipse hyperbola parabola

Parametric curve (circle)

• The simplest non-linear curve - circle

- circle with radius R centered at the origin

• x(t) = R cos(2t)

• y(t) = R sin(2t)

• 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

Circular arc

• Circle with center at (xc, yc)

• x(t) = R cos(2t) + xc,

• y(t) = R sin(2t) + yc ,

Parametric curve

• Ellipse– x(t) = a cos(2t)– y(t) = b sin(2t)

• Hyperbola– x(t) = a sec(t)– y(t) = b tan(t)

• parabola– x(t) = at2

– y(t) = 2at

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)

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

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

• Tangent line– The line that contains the 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 vector

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(2t), R sin(2t)) , 0 t 1

Find tangent vector at t and tangent line at F(t).

example

• Answer

dx = Rcos(2t), dy = R sin(2t)

x’(t) = dx/dt = - 2 Rsin (2t),

y’(t) = dy/dt = 2Rcos(2t)

Tangent vector = (- 2 Rsin (2t), 2Rcos(2t))

example

• Answer

• Tangent line– F(t) + u(F’(t))– (Rcos(2t), R sin(2t)) + u(- 2 Rsin (2t),

2Rcos(2t))– (Rcos(2t) + u(- 2 Rsin(2t))) , (R sin(2t) +

u(2Rcos(2t)))

Example • Check / prove

• Let say, t = 0,Tangent vector = (- 2 Rsin (2t), 2Rcos(2t))

= (0, 2R)

tangent line = (Rcos(2t) + u(- 2 Rsin(2t))) , (R sin(2t) +

u(2Rcos(2t)))

= (R, u(2R))

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)

• 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 Straight line curvature = ? 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.

• 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.

Curve use in design

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

• Interactive design consists of the following steps

1. Lay down the initial control points

2. Use the algorithm to generate the curve

3. If the curve satisfactory, stop.

4. Adjust some control points

5. Go to step 2.

An interactive curve design

basic curve surface

y t parameter t

x t yt

point x t

parametric line example

a14 circle t

parametric line answer

r sin2 t xt

parametric line positions

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