CS 450: COMPUTER GRAPHICS

REVIEW: DRAWING ELLIPSES AND OTHER CURVES

SPRING 2015

DR. MICHAEL J. REALE

OVERVIEW OF DRAWING CURVES

• Explicit form y = f(x)

• Parametric form y = f(s), x = g(s)

• Implicit form F(x,y) = 0

• Two approaches:

• Use explicit or parametric form approximate curve with line segments

• Parametric form evenly-spaced points along curve path

• Can connect points OR use best-fitting line

• Use implicit form derive incremental midpoint method

• Take advantage of any curve symmetry

ELLIPSES• Symmetric only for quadrants

• Circles symmetric in all octants

• Equations:

• Two foci for point on ellipse, sum of distances from foci is constant:

• General ellipse equation:

• Axis-aligned (standard) ellipse:

• Two axes:

• Major axis = line passing through both foci

• Minor axis = line perpendicular to major axis

• Semimajor/semiminor axes radii aligned with major/minor axes, respectively

cdd 21

http://commons.wikimedia.org/wiki/File:Drawing_an_ellipse_via_two_tacks_a_loop_and_a_pen.jpg#mediaviewer/File:Drawing_an_ellipse_via_two_tacks_a_loop_and_a_pen.jpg

022 FEyDxCxyByAx

1

22

y

c

x

c

r

yy

r

xx

MIDPOINT ELLIPSE ALGORITHM

• Align ellipse with x/y axes

• Start at (0,ry )

• Use two separate midpoint algorithms:

• 1) Increment x choose y UNTIL slope < -1.0

• 2) Then, decrement y choose x

• Use symmetry to copy points to other 3 quadrants

IMPLICIT ELLIPSE FORMULA

0),( 222222 yxxy rryrxryxF

boundary ellipse theOUTSIDE is y)(x, if 0,

boundary ellipse theON is y)(x, if 0,

boundary ellipse theINSIDE is y)(x, if 0,

),( yxF

…which has the following properties:

NONSTANDARD ELLIPSES

• For ellipses not aligned with coordinate axes:

• 1) Adapt formulas to work over ENTIRE ellipse path

• 2) Use standard ellipse use transformation to rotate ellipse points

MIDPOINT ELLIPSE ALGORITHM: REGION 1

• 1. Input rx, ry, and ellipse center (xc, yc); get first point for ellipse centered on origin:

• 2. Get initial decision variable for region 1:

• 3. While (2ry2x < 2rx

2y)

• If p1k < 0 choose (xk + 1, yk)

• Otherwise choose (xk + 1, yk – 1)

),0(),( 00 yryx

2220 4

11 xyxy rrrrp

21

21 211 ykykk rxrpp

122

12

1 2211 kxykykk yrrxrpp

MIDPOINT ELLIPSE ALGORITHM: REGION 2

• 4. Get initial value of decision variable for region 2 from last position (x0, y0) calculated in region 1:

• 5. While (y != 0)

• If p2k > 0 choose (xk, yk – 1)

• Otherwise choose (xk + 1, yk – 1)

• 6. For both regions, define symmetry points: (x,y), (-x,y), (x,-y), and (-x,-y)

• 7. For each position (x,y), plot:

2220

22

02

0 12

12 yxxy rryrxrp

21

21 222 xkxkk ryrpp

21

21

21 2222 ykxkxkk rxryrpp

c

c

yyy

xxx

CONIC SECTIONS

• Equation:

• Intersection of plane with double cone

• Does not include “degenerate” conics point, line

022 FEyDxCxyByAx

http://mathworld.wolfram.com/ConicSection.html

hyperbola ,0

parabola ,0

circle)(or ellipse ,0

42 ACB

DRAWING CONIC SECTIONS

• Use symmetry:

• Circles across 8 octants

• Ellipses/Hyperbolas across 4 quadrants

• Parabolas across vertical axis

• Compute decision variable formulas for areas where slope is less than 1.0 and greater than 1.0

• Go through midpoint algorithm, checking when to switch because of slope

POLYNOMIALS AND SPLINE CURVES

• Use parametric form

• Example:

• Parameter u [0,1] start/end of curve

• Approximate with line segments (small increments of u)

• Spline curve (splines) = continuous curve formed with polynomial pieces

33

2210

33

2210

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