TNM079 Modeling & Animation Lectures

Flashcards based on lecture notes.

Vertex

A vertex with the coordinates x, y, z.

Face

Contains pointers to the vertices in a vertex list.

Indexed face list

Stores each vertex only once.

Half-edge mesh

Combines face information with neighbor information.

Manifold mesh

Each edge is shared by two triangles.

vert

Half edge data structure element that contains info about the adjacent vertex

Next

Half edge data structure element that contains info about the next half-edge

Prev

Half edge data structure element that contains info about the previous half-edge

Pair

Half edge data structure element that contains info about paired half-edge

Face

Half edge data structure element that contains info about the adjacent face

1-ring of a vertex

The ensemble of triangles sharing a vertex v

Euler-Poincaré formula

A relation between the number of edges, vertices, faces, loops, shells and genus in a manifold mesh.

Genus

The number of topological holes.

Osculating circle

Approximates the curve in a small segment that contains the point P.

Curvature

A measure of how much a curve deviates from being a straight line.

Principal curvatures

The minimum and maximum curvature at a point on a smooth surface.

Gaussian curvature

The product of the two principal curvatures.

Mean curvature

The average of the two principal curvatures.

Edge collapse

Merging two neighboring vertices to one.

Vertex split

Creating a new pair of vertices.

Static simplification

Level of detail is precomputed offline for a few cases.

Dynamic simplification

Level of detail can change continuously at run time.

View dependent simplification

Dynamic simplification that can vary across the object.

Vertex merging

Does not always remove an edge, can also connect nearby objects. It can lead to non-manifold meshes.

Quadric error metric

The distance between a new vertex position and the original surface.

Original vertex

Vertex of the mesh given in homogeneous coordinates .

Error Metric

A scalar number that quantifies the error introduced by moving an original vertex to a new position.

Kpi

A 4 × 4 matrix used in calculating the quadratic form.

Quadratic form

Quantifies the error we get when we move V1 and a second one quantifies that from moving V2.

Bézier spline

An approximation of a curve with start point, end point and handles.

Control points

The start point, end point, and handles of a curve.

Linear interpolation

The easiest interpolating function involves control points Ci and Ci+1 next to each other along the curve.

Spline curve

Approximated by a sequence of spline segments.

Spline base

A curve has one maximum value at some point and converge to zero as we move away. Functions with local support.

Knot

Curve parameter.

Cardinal spline base

Equal distances between ti give a …

B-spline

Use the global parameter t and not one s per spline segment.

knot index

B-spline parameter

Splines

Relates the vertices of a mesh to control points of a spline surface.

Curve refinement

Doubling the number of the knots for a given curve doubles the control points. Same curve, smoother convex hull.

Control polygon

A height field in t, u space if cij is a scalar and a quad mesh if cij is a 3D vertex.

Loop’s algorithm

An algorithm that refines a triangle mesh.

Implicit surface

Function of the position x = (x, y, z).

Isosurface

Surface with function constant.

Manifold Surface

Have no holes or self-intersections

Standard convention (Implicit surfaces)

Zero-contour separates exterior points x with F(x) > 0 from interior ones with F(x) < 0.

Gradient vector

Points along the direction where F(x) increases fastest

Finite difference scheme

Small but finite ∆x gives a…

Ray tracing

Used to find intersections between rays and complex implicit surfaces.

Marching cubes

A method involving subdividing space into voxels to render implicit surfaces

Quadrics

Implicit functions that involve polynomials up to the maximum order of 2 (quadratic)

Level set

Function is time-dependent.

Level Set F(x,t) = 0

The boundary (contour or surface) of the Level Set .

Signed distance function

Function fulfills | |∇F| | = 1.

Advection

Transporting information in the direction set by the speed vx

Upwind scheme

If vx > 0, the information at the cell xi comes from the left: xi−1.

Courant-Friedrichs-Levy (CFL) number

Determines the stability of a numerical scheme and depends on its details.

Compact Support

Smooth B-spline that is not zero outside a local region.

Sampling the fourth-order B-spline

Approximating the Level set at ts = s∆t = 0.01 (400 samples).

Projection operator

We replace the linear term by a projection operator P, which enforces incompressibility ∇ · u = 0.

Operator splitting

A numerical method for solving differential equations where you apply forces one by one.