Barycentrc Coordnates From: Mean Value Coordnates for Closed Trangular Meshes by Ju et al.
Motvaton Data nterpolaton from the vertces of a boundary polygon to ts nteror Boundary value problems Shadng Space deformatons Parametrzaton
Barycentrc Coordnates n Trangles A pont v s the weghted barycenter n a trangle {v 1, v 2, v 3 }, wth weghts {w 1, w 2, w 3 } f: v w v w v w v 1 1 2 2 3 3 w w w 1 2 3 The weghts are called barycentrc coordnates (Möbus, 1827, Der Barycentrsche Calcul)
Barycentrc Coordnates n Trangles The barycentrc coordnates n a trangle are unque Up to a constant scalng factor Normalzed (unque) barycentrc coordnates: 3 w 1 w
Barycentrc coordnates n Trangles Propertes: Lagrange: ( v j ) j Postvty: 0 Lnearty Lnear Interpolaton of data: 3 f ( v) ( v) f 1 How do barycentrc coordnates generalze to general polygons?
Generalzed Barycentrc Coordnates Homogeneous coordnates w : w () v 1 Generalzed coordnates: 1 Propertes: Partton of unty: v Reproducton: j n 1 n () v w () v v n w () v n 1 j w() v Lnear Reproducton (for lnear ф) n 1 n 1 ( v) ( v) ( v) 1 () v v v
Generalzed Barycentrc Coordnates Not unque. Many types proposed. We wll study three man coordnates: Wachspress Harmonc Mean value
Wachspress Coordnates Only apply to convex polygons Three ponts constructon: w A A B 1 A 1 A The areas are sgned!
Wachspress Coordnates Rephrasng the expresson: A A B sn( ) v v v v 1 1 1 1 2 AA 1 sn( 1) v v 1 v p sn( ) v v 1 sn( 1) cot( 1) cot( ) 2 2 sn( )sn( ) v p v p 1 Every coordnate depends on the vertex ts neghbors Such coordnates are called threepont coordnates.
Wachspress Interpolaton It s evdent that nonconvex regons have poles
Wachspress Coordnates Not well defned on concave polygons Negatve weghts, whch mght lead to poles (zero weght sum). Not defned on the boundary Lmt exsts.
Harmonc Coordnates Assgn 1 to vertex v, and 0 to all other vertces Weght vary lnearly on edges Solve Laplace equaton nsde polygon: 0 ( ) v j j
Harmonc Coordnates Well-defned and postve for arbtrarly-defned polygons Laplace equaton lnear nteror s the harmonc nterpolant of boundary values. Lnear precson Man setback no closed form Hard to compute Usually solved by dscretzaton
Mean Value Coordnates A closed-form approxmaton of harmonc nterpolaton. Harmonc functon hold the mean-value property: 1 f ( x) f ( t) dt C For crcles around the pont x. C
Mean-Value Interpolant Idea - project the boundary curve unto a unt sphere around the pont Integrate by projecton angles, wth respect to projecton dstance: f( x) 2 0 2 0 f ( C( t)) d( C( t)) 1 d( C( t)) d d
Dscrete Boundary Boundary s pecewse-lnear (Polygon) The ntegral over one edge [P 0,P 1 ] becomes: 2 1 f( C( )) f( p0 ) f( p1 ) 1 0 d tan d( C( )) d( p0, x) d( p1, x) 2 Rearrangng terms to be per-vertex we get: w( x) 1 tan tan 2 2 v x
Mean Value Coordnates Defned anywhere n the plane
Dscrete Harmonc coordnates MVC - three-pont coordnates as well Another member of ths famly: Dscrete Harmonc coordnates (cotan weghts) w( x) cot( a) cot( b) 2 DHC can also have poles
Comparson Three Pont Coordnates
Applcatons: Space Deformaton Each nternal pont x has a set of barycentrc coordnates Old postons (vector data) of boundary: Lnear precson: nternal ponts are reproduced: x 1 New postons: 1 n v 1.. n n n 1 { u u } y u v 1.. vn
Applcatons: Vector Feld Interpolaton
Applcatons: Image Warpng
Applcatons: Parametrzaton Usng the barycentrc coordnates n each one-rng as edge weghts Postonng every pont as barycenter of ts 1-rng wth the weghts. Exercse: Dsk parametrzaton wth Unform Dscrete harmonc coordnates (lecture) Mean-value coordnates
Extensons 3D\nD coordnates Non affne-nvarant coordnates Green Coordnates Complex Coordnates