Geometric Modeling of Points, Lines, and Curves

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Geometric Modeling of Points, Lines, and Curves

• Dr. Jack G. Zhou
• Mechanical Engineering and Mechanics Department
• Drexel University

2
First Words

• Elements of geometric modeling frames, points,
  lines, curves, surfaces, and solids
• Evolution of geometric modeling
• 2D drafting
• 3D wireframe modeling
• surface modeling
• solid modeling
• Wireframe models consist of points, lines, and
  curves

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Contents

• Visualization of geometric models
• Definition and storage of point entities
• Definition and storage of line entities
• Definition and storage of circles and arcs
• Definition and storage of parametric cubic spline
  curves
• Definition and storage of Bezier curves

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Visualization of geometric models

• The user visualizes the space of the world
  coordinate frame through a visualization window
  defined within the world frame
• A CAD/CAM systems graphical user interface (GUI)
  presents the image within the visualization
  window to the user via a screen window defined on
  a computers display monitor
• A GUI allows the user to visually travel through
  the world space by relocating the visualization
  window, via a mouse, tablet, joystick, or virtual
  reality device

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The image on a screen window

• A visualization window is typically defined as an
  X-Y rectangular region in the XY-plane of a
  visualization frame
• The image of a geometric object may be the
  orthographic projection of the object onto the
  visualization window
• Only the image inside the visualization window
  needs to be revealed on the screen window

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Definition and storage of point entities

• Each point may reside directly in the world frame
  or in a local frame
• Either static or dynamically allocated arrays can
  be used to store the relevant (x, y, z)
  coordinates of point entities
• A point may be defined in many ways coordinate
  entry via keyboard or mouse, end of a line or
  curve, intersection of two curves, etc.

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Definition and storage of line entities

• Each line may reside directly in the world frame
  or in a local frame
• A line can be stored as the point set between two
  point entities
• A line may be defined in many ways end-point
  coordinate entry via keyboard or mouse, tangent
  between a point and a curve, tangent between two
  curves, etc.

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Analytical equations of a line
and
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Parametric equations of a line

• The same line can be represented in a vector form
  with one real-number parameter ?
• p(?) (1- ??p1 p2, - ??lt ? lt ??
• Note that the finite parametric range -1 ??? ??1
  correlates to the line segment between p1 and p2.

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Intersection of two lines

• The intersection of two given lines
• p(?) b1 ??d1
• and p(?) b2 ??d2
• would be a point which satisfies both parametric
• equations. Thus finding the intersection is
  equivalent
• to finding the values of ? and ? from the
  following
• equation
• b1 ??d1 b2 ??d2

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Basic curves

• Circles and arcs
• Parametric cubic spline curves
• Bezier curves

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Definition and storage of circles and arcs
Y
Z
Z
a
Y
b
X
X

• The analytical equation of a circle centered at
  (xc, yc) and
• having a radius of r, in an XY-plane, is
• (x - xc)2 (y - yc)2 r2
• The same circle can be expressed in the following
  parametric
• form
• x xc r cos ?
• and y yc r sin ?
• with - ? lt ? ? ?

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Representing a circle or arc with three points

• 1) In an XY-plane, a set of three points center
  (C), start (S), and terminate (T) can be used to
  represent an arc
• 2) For a circle, S T
• 3) The defining frame
• 4) The Z-coordinate

S
C
Y
T
X
C
T
S
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Definition and storage of parametric cubic spline
curves

• A parametric cubic curve segment can be traced
• through parameter u as defined in its algebraic
• form
• x(u) a3x u3 a2x u2 a1x u a0x
• y(u) a3y u3 a2y u2 a1y u a0y
• z(u) a3z u3 a2z u2 a1z u a0z
• where 0 ? u ? 1
• or in vector notation
• p(u) a3 u3 a2 u2 a1 u a0

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Shaping a parametric cubic curve via ai

• the algebraic form is not intuitive
• ai offers little visual clue in the shape of a pc
  curve
• an equivalent geometric form can be obtained
  through the following geometric boundary
  conditions
• p(0) a0
• p(1) a0 a1 a2 a3
• pu(0) a1
• pu(1) a1 2a2 3a3

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Shaping a parametric cubic curve via p and pu

• After solving the set of four simultaneous
  equations in four
• unknowns, we obtain
• a0 p(0)
• a1 pu(0)
• a2 - 3p(0) 3p(1) - 2 pu(0) - pu(1)
• a3 2p(0) - 2p(1) pu(0) pu(1)
• Thus
• p(u) (2u3 - 3u2 1) p(0) (-2u3 3u2) p(1)
  
• (u3 - 2u2 u) pu(0) (u3 - u2) pu(1)

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The geometric form reveals the constraints on the
curve
pu(0)
p(0)
p(1)
pu(1)

• The shape of the cubic curve is determined by
  summing the
• products of four cubic polynomial functions with
  the four
• geometric coefficients shown above. The curve
  passes
• through the two end points and follows the two
  end-tangent
• vectors. The four polynomials are known as
  blending
• functions. The visual connection is evident.

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The blending functions of a parametric cubic
curve

• Define blending functions as
• F1(u) 2u3 - 3u2 1
• F2(u) -2u3 3u2
• F3(u) u3 - 2u2 u
• F4(u) u3 - u2
• and the geometric form simplifies to
• p(u) F1(u) p(0) F2(u) p(1) F3(u) pu(0)
  F4(u) pu(1)

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Definition and storage of Bezier curves

• A general Bezier curve is formed by the following
  polynomial
• blending
• n
• p(u) ? pi Bi,n(u) u ? 0, 1
• i0
• where the Bernstein polynomials
• Bi,n(u) C(n, i) ui (1 - u)n-i
• and where the binomial coefficient
• C(n, i) n!/(i! (n-i)!)
• A nth-degree Bezier curve is shaped by a
  characteristic or control
• polygon formed by n1 control points.

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Cubic Bezier curve
p1
p0
p2
p3

• A cubic Bezier curve is shaped by four control
  points pi, and
• p(u) (1 - u)3p0 3u(1 - u)2p1 3u2(1 - u)p2
  u3p3
• The curve passes through the two end points and
  follows the
• tangent directions determined by the first and
  last edges of
• the control polygon. It is equivalent to the
  parametric cubic
• curve when
• pu(0) 3(p1 - p0) and pu(1) 3(p3-p2)

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Concluding remarks

• A GUI is a prerequisite to interactive geometric
  modeling
• Points, lines, and curves are the basic elements
  of wireframe models