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Course13
CurvesandSurfaces
Course13CurvesandSurface-----SurfaceRepresentationRepresentationInterpolationApproximation-----SurfaceSegmentationSegmentationRegistration
BasicGeometry1)?Curve:Parameterform:e.g.,alinesegmentbetweenand:?Implicitform: Explicitform:
2)?Surface:Parameterform:Implicitform:Explicitform: (graphsurface)
3)Planes:Assumeare3non-collinearpointsinspace.Aplaneisuniquelydeterminedbythepoints.LetTheequationoftheplaneis
whereisthenormaloftheplane.Inparametricformoftheplane,wecanredefineThentheplaneequationis:whereareparameters,
4)SurfaceCurvatures:AssumeasurfaceinformofThenormalofthesurfaceofpointiswherearepartialderivativesofthesurface.
(1)??Normalcurvature:TheintersectionofasurfaceandaplanethatpassesthroughthenormalatapointPofthesurfaceisa2Dcurve.Thecurvatureofthe2DcurveatpointPiscallednormalcurvatureofthesurfaceatpointP.----Atasurfacepoint.Thesurfacehasinfinitemanynormalcurvatures.----Theminimumandmaximumnormalcurvatures,K1andK2arecalledprincipalcurvatures.
(2)GaussianandmeancurvaturesGaussiancurvature: Meancurvature:
(3)TypesofasurfaceregionK(x,y)??+0–???+peakridgesaddleridge0?flatminimalsurface–pitvalleysaddleValley
(4)CurvatureCalculationForasurface
2.Representationofcurveandsurface1)?Curverepresentations:-----Piecewisecurvesegments-----Jointcondition:smoothatleastinthefirstorderderivatives,i.e.continuousincurveslope.-----Lowerestorderofpolynomials.Thisleadstocubicsplinecurve:x(t)=axt3+bxt2+cxt+dxy(t)=ayt3+byt2+cyt+dyz(t)=azt3+bzt2+czt+dz
Whenoneuses: thenacurvecanbewrittenas:
Foracurve,itisinpiecewiseform:Withconnectionconditions:
2)Polygonalmesh
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