Admin; Starter CodeAdmin; Starter Code

• Website Ready: Plans evolving...Website Ready: Plans evolving...

• Starter Code Ready: also evolves...Starter Code Ready: also evolves...

• About Your Projects: About Your Projects: – Simple directory-based version control: Simple directory-based version control: – Save & NumberSave & Number everything that works: everything that works:

NEVERNEVER edit an unsaved working program edit an unsaved working program– Always, Always: One Little Step at a Time!Always, Always: One Little Step at a Time!– Explain it as you write it (comments)Explain it as you write it (comments)

“Writing IS thinking”“Writing IS thinking”

Admin; Starter CodeAdmin; Starter Code

• Website Ready: Plans evolving...Website Ready: Plans evolving...• Starter Code Ready: also evolves...Starter Code Ready: also evolves...

• About Your Projects: About Your Projects: – Simple directory-based version control:Simple directory-based version control:– What works well for me:What works well for me:

• One master directory for the projectOne master directory for the project• All files of latest version in All files of latest version in oneone directory directory• Numbered previous versions: ‘snapshot’ copiesNumbered previous versions: ‘snapshot’ copies• Copy master directory EVERY time something works.Copy master directory EVERY time something works.• Program is ALWAYS working; one step at a time...Program is ALWAYS working; one step at a time...• > 1 hour without compile/run? too big a step!> 1 hour without compile/run? too big a step!

ENDEND--Please download the starter code, compile and run.--Please download the starter code, compile and run.--Then you can get started on Project 1 right away.--Then you can get started on Project 1 right away.

Conic MethodsConic Methods

• Review conics first Review conics first (pg.8)(pg.8)

– ‘‘Conics’Conics’ == intersection of cone & plane: == intersection of cone & plane:– Many possible shapes:Many possible shapes:

circles, ellipses, parabola, hyperbola, circles, ellipses, parabola, hyperbola, degenerates (lines & pointsdegenerates (lines & points

Conic MethodsConic Methods

– Equation of any/all conics solve a 2D quadratic: Equation of any/all conics solve a 2D quadratic: a axx22 + b + bxyxy + c + cyy22 +d +dxx +e +eyy +f = 0 +f = 0

– Write in homogeneous coordinates:Write in homogeneous coordinates:

xxTTCx = 0Cx = 0– C is symmetric, C is symmetric, 55DOF (DOF (not 6,not 6, because because xx33 scaling) scaling)

– Find any Find any CC from 5 homogeneous points from 5 homogeneous points(solve for null space—see book (solve for null space—see book

pg 9)pg 9)

aa b/2 d/2 b/2 d/2

b/2 c e/2 b/2 c e/2 d/2 e/2d/2 e/2 f f

xx11

xx22

xx33

= 0 = 0 xx11 x x22 x x33

a ‘Point Conic’a ‘Point Conic’

Conic MethodsConic Methods

– Matrix C makes conics from points: Matrix C makes conics from points:

xxTTCx = 0Cx = 0 CC is a is a ‘point conic’‘point conic’

– Given a point Given a point xx on a conic curve, the on a conic curve, the homog. tangent line homog. tangent line ll is given by is given by l = C xl = C x

– Matrix C* makes conics from lines:Matrix C* makes conics from lines:

llTTC*l = 0C*l = 0 C*C* is a is a ‘Dual Conic’‘Dual Conic’

defined by tangent linesdefined by tangent lines ll instead of points.instead of points.

Conic MethodsConic Methods

– If If C C is non-singular (rank 3), then is non-singular (rank 3), then C* = CC* = C-1-1

– If If C C (or (or C*C*) has…) has…

Rank 3: it is an ellipse, circle, parab., hyperb.Rank 3: it is an ellipse, circle, parab., hyperb.

Rank 2: it is a pair of lines (forms an ‘x’)Rank 2: it is a pair of lines (forms an ‘x’)

THE BIG IDEA---MOST USEFUL PROPERTYTHE BIG IDEA---MOST USEFUL PROPERTYProjective transform of a conic Projective transform of a conic CC is conic is conic C’C’::

C’ = HC’ = H-T -T C HC H-1-1