Geometric Modeling: Fundamentals

1
Geometric Modeling Fundamentals

• Ability to represent points in computer
• Ability to transform points and lines
• Points and lines represent objects
• Computer hardware/software to display objects
• It is necessary to scale, rotate, tranlate,
  stretch or develop perspective views

2
Point Representation

• Points in 2D x y Points in 3D x y z
• Such matrices are called VECTORS
• A set of points, each of which is a position
  vector with respect to a local coordinate system
  as a matrix of numbers in computer
• Position of points can be controlled by changing
  the matrix
• Display can be made with lines, curves, surfaces,
  volume display etc.,

3
Transforms and Matrices

• Many physical problems lead to matrix formulation
  ( number store, a network, the coefficients of a
  set of equations)
• AT B (given matrices A and B, solve for T)
• T A-1 B (A inverse B)
• B can be computed given A and T. Geometrical
  Operator (matrix multiplication)

4
Point Transformation

• Point P transformed using a 2D transform
• x y a b axcy bxdy x y
• c d
• Consider ad1 and bc0 identity matrix. No
  change
• Set bc0, d1 Result ax y. Scale change in x.
  Stretching in x direction only

5
Point Transformation

• Let bc0. Result ax dy. Stretching in both x
  and y directions
• If adgt1, enlargement of coordinates
• If adlt1, compression of coordinates with respect
  to the origin
• If a and/or d is negative, result is REFLECTION.
  Set bc0, d1, a-1. Result -x y. Reflection
  about y-axis.

6
Point Transformation

• Reflection about x-axis results by setting bc0,
  a1, d-1 Result x y
• Practical cases using reflection about the
  ORIGIN bc0, ad lt 0 Result -x -y
• NOTICE reflection, stretching and scaling using
  only the DIAGONAL terms of the matrix

7
Point Transformation

• SHEAR effect is of interest Set ad1 and c0
  Result x bxy x y . Shear in Y using
  ad1 and b0 Result cyx, y
• 2D transform applied to the ORIGIN of the
  coordinate system ??
• ORIGIN invariant under general 2x2
  transformation. LIMITATION. Use HOMOGENEOUS
  coordinates

8
Line Transformation

• Use of two position vectors which are end points
  of a straight line
• Position and orientation can be controlled
• Operation of drawing will be dealt with under
  visualization using hardware/software
• Consider a transformation matrix T 1 2
• Line A 0 1 to B 2 3 3
  1

9
Line Transformation

• Shear effect on point 0 1
• AT 3 1 and BT 11 7
• AB is same as L 0 1
• 2 3
• LT is now 3 1
• 11 7
• Increased length of line, orientation change

10
Midpoint Transformation

• Line yx1 is transformed into
• Line y (3/4)x 5/4
• So any 2x2 matrix transforms ANY straight line
  into another straight line
• Proof of midpoint transformation is confirmed by
  transforming the midpoint of first line. All
  points on first to second line.

11
Parallel lines

• AB and EF are parallel lines
• Consider slope of AB, EF
• M1 (y2 y1) / (x2 x1)
• Transform the lines and see the slope reduce to
  m2 (b dm1) / (a cm1)
• Independent of x1, x2, y1, y2 and m, a,b, c, d
  are same for AB and EF.
• Parallelograms remain as parallelograms

12
Intersecting Lines

• AB and EF are intersecting lines (see figure 2-3
  in notes)
• Initial point of intersection transforms to the
  new point of intersection of the transformed
  lines
• Rotation, Reflection and Scaling
• Transform used 1 2
• 1 3

13
ROTATION

• Consider a simple traiangle ABC
• Rotation by 90 degrees Counter-Clockwise about
  the ORIGIN
• 3 -1 0 1 1 3 90 degrees CCW
• 4 1 -1 0 -1 4
• 2 1 -1 2

14
Rotation

• 180 degree rotation about ORIGIN CCW is produced
  using -1 0
• 0 -1
• 270 degree rotation is created by using
• 0 -1 about ORIGIN
• 1 0
• No SCALING, No REFLECTION occured

15
Reflection

• Pure 2D rotation in xy-plane occurs about an axis
  NORMAL to the xy-plane
• REFLECTION is a 180-degree rotation about an axis
  IN THE xy-plane.
• The same triangle with 2 Reflections are shown in
  the figure 2-4b (see notes)

16
Reflection

• Reflection about yx axis in the xy-plane occurs
  using 0 1
• 1 0
• A reflection about y0 occurs using 1 0
• 
  0 -1

17
Scaling

• Scaling is controlled by the magnitude of the two
  terms in the primary DIAGONAL of the T matrix
• Using 2 0
• 0 2 a 2 times magnification about
  origin occurs
• Unequal scale factors create distorsion
• Plane surfaces can be easily transformed

18
Combined Transformations

• Typically more than one transform used
• Controls shape and position
• NON-COMMUTATIVE
• Order is important in matrix multiplications
• Consider a 90-degree rotation and then reflection
  on one of the vertices x y of a triangle

19
Combined Transformations

• First rotate and then apply reflection
• Result y x
• Now first apply reflection and then rotate
• Result -y -x
• NOT THE SAME. NON-COMMUTATIVE
• Mapping from one plane into a second plane