Embed Size (px)
Citation preview
Basic Geometric Processing
Spaces 1D, 2D, and 3DPoint
Line
Plan
),,(),,,(),,,(
),(),,(),,(
,,
222111000
221100
210
zyxzyxzyx
yxyxyx
xxx
),,(),,(),( zyxyxx
),,(),,,(
,),(),,(,,
111000
110010
zyxzyx
yxyxxx
Points/VectorsA point p = (x,y,z) is
also the vector pVector Operation
The Length of the vector v
Normalization
Dot product of u and v
222
22
|),,(|||
|),(|||
zyxzyxv
yxyxv
)),((||||||
),,(),,( 111000
vuCosvuvu
zyxzyxvu
||/ˆ vvv u
vu+v
Vector OperationCross Product
Courtsy:Wikipedia
),,(
)()()(
vuvuvuvuvuvu
vuvuvuvuvuvu
vvv
uuu
xyyxxzzxyzzy
xyyxkxzzxjyzzyi
zyx
zyx
kji
vu
LinesEquation
Parametric representation
baxy
cbyax
0
vtppt 0 v0p
vtvtp 0
PlanPlan Normal
Plan Equation
),,( nnn zyxN ),,( nnn zyxN
0
0),,(
dzzyyxx
dNzyx
nnn
Point, Line, and PlanPoint-line relation
Inclusion, above/belowPoint-plan relation
Inclusion, above/belowLine-plan relation
Intersection, inclusion, parallel, above/belowPlan-plan relation
Intersection, inclusion, parallel, above/below
Polyline and PolygonVertices and PolylineConvex and Concave PolygonsSimple and Complex PolygonsPolygon Triangulation Inside and outside a PolygonConvex Hull
