Geometric Modeling Fundamentals

Geometry and topology

Geometry is the metric information of the model: actual coordinates of points, equations of curves and surfaces, lengths, angles, radii. Topology is the connectivity information: how vertices, edges and faces are joined, and which side of a face is material. It does not change when the shape is stretched.

Euler-Poincare formula

For a valid solid with holes (genus) and inner loops:

$$V - E + F - R = 2\,(S - H)$$

where $V$ = vertices, $E$ = edges, $F$ = faces, $R$ = rings (inner loops on faces), $S$ = shells, $H$ = through-holes (genus).

Numerical

Given: $V = 16$, $E = 24$, $F = 10$, one shell $S=1$, one through-hole $H=1$.

$$\begin{aligned} 16 - 24 + 10 - R &= 2(1-1) = 0 \\ 2 - R &= 0 \\ R &= 2 \end{aligned}$$

The two rings are the square hole boundaries on the top face and on the bottom face. Each of these faces has an outer loop and one inner loop, so $R = 2$, which agrees with the physical block. The formula is satisfied, so the topology is valid.

Answer: $R = 2$ inner loops (top and bottom faces); $V-E+F-R = 0 = 2(S-H)$, so the solid is valid.

Coordinate systems

• Cartesian $(x, y, z)$: three mutually perpendicular axes; simplest for prismatic shapes and used internally by CAD.
• Cylindrical $(\rho, \phi, z)$: radial distance in the XY plane, azimuth angle from +X, and height; suited to shafts, holes and revolved parts.
• Spherical $(r, \theta, \phi)$: distance from origin, polar angle from +Z, azimuth from +X; suited to spheres and radial scanning.

Relations:

$$\rho=\sqrt{x^2+y^2},\quad \phi=\tan^{-1}\frac{y}{x},\quad r=\sqrt{x^2+y^2+z^2},\quad \theta=\cos^{-1}\frac{z}{r}$$ $$x=r\sin\theta\cos\phi,\quad y=r\sin\theta\sin\phi,\quad z=r\cos\theta$$

(a) Point (3, 4, 5)

$$\begin{aligned} \rho &= \sqrt{3^2+4^2} = 5 \\ \phi &= \tan^{-1}(4/3) = 53.13^\circ \\ r &= \sqrt{9+16+25} = 7.071 \\ \theta &= \cos^{-1}(5/7.071) = 45^\circ \end{aligned}$$

Cylindrical: $(5,\ 53.13^\circ,\ 5)$. Spherical: $(7.071,\ 45^\circ,\ 53.13^\circ)$.

(b) Spherical to Cartesian

$$\begin{aligned} x &= 10\sin 60^\circ\cos 45^\circ = 10(0.8660)(0.7071) = 6.124 \\ y &= 10\sin 60^\circ\sin 45^\circ = 6.124 \\ z &= 10\cos 60^\circ = 5.000 \end{aligned}$$

Answer: (a) cylindrical (5, 53.13 deg, 5); spherical (7.071, 45 deg, 53.13 deg). (b) Cartesian (6.124, 6.124, 5.000) in the same length units.

Given

$\mathbf A = (2, 3, 1)$, $\mathbf B = (4, -1, 2)$.

(a) Dot product

$$\mathbf A\cdot\mathbf B = 2(4)+3(-1)+1(2) = 8-3+2 = 7$$

(b) Cross product

$$\mathbf A\times\mathbf B= \begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\2&3&1\\4&-1&2\end{vmatrix} = (6+1)\mathbf i-(4-4)\mathbf j+(-2-12)\mathbf k = 7\mathbf i+0\mathbf j-14\mathbf k$$

(c) Angle

$|\mathbf A|=\sqrt{14}=3.742$, $|\mathbf B|=\sqrt{21}=4.583$.

$$\cos\theta=\frac{7}{3.742\times4.583}=0.4082,\qquad \theta=65.9^\circ$$

(d) Unit normal

$|\mathbf A\times\mathbf B|=\sqrt{49+0+196}=15.652$

$$\hat{\mathbf n}=\frac{(7,0,-14)}{15.652}=(0.447,\ 0,\ -0.894)$$

(e) Area of parallelogram

Area $=|\mathbf A\times\mathbf B| = 15.65$ square units.

Uses in modelling

• Dot product: angle between edges, projection of a point on a line, checking perpendicularity (dot product = 0).
• Cross product: surface normal of a face (used in shading, offsetting, STL facet normals), area of a face, and the sign of a turn.

Answer: A.B = 7; AxB = (7, 0, -14); angle = 65.9 deg; unit normal = (0.447, 0, -0.894); area = 15.65 square units.

Non-parametric representation gives a direct relation between coordinates: explicit $y=f(x)$ or implicit $f(x,y)=0$. Parametric representation expresses each coordinate as a function of an independent parameter $u$: $x=x(u)$, $y=y(u)$, $z=z(u)$, with $u$ in a range, usually $0\le u\le1$.

Why parametric is preferred

1. Curves can be closed, multi-valued or have vertical tangents.
2. Easy to generate points at equal steps of $u$ for display and tool paths.
3. Tangent vector is simply $d\mathbf P/du$.
4. Segments are bounded and can be joined with continuity conditions.
5. Same equation works in 2D and 3D, and with any axes.

Circle of radius 5

Non-parametric (implicit): $x^2+y^2=25$; explicit form needs two branches $y=\pm\sqrt{25-x^2}$.

Parametric:

$$x = 5\cos\theta,\qquad y = 5\sin\theta,\qquad 0\le\theta\le 2\pi$$

For example $\theta=90^\circ$ gives $(0,5)$ directly, where the explicit form has an infinite slope.

Geometric modelling is the computer representation of the shape of an object. Three types are used, in increasing order of completeness.

Wireframe modelling

The object is represented by its edges (lines, arcs, curves) and vertices only; there are no faces.

• Advantages: small data, fast to create and display, little memory, good for simple layouts and tool-path skeletons.
• Limitations: ambiguous (a cube can be read in several ways), no hidden-line removal, no surface or volume data, no mass properties, no shading.

Surface modelling

The object is described by its bounding surfaces (plane, ruled, revolved, Bezier, B-spline, Coons patches), stored with edges.

• Advantages: can model free-form shapes, supports hidden-line removal and shading, gives surface normals for NC machining and for aerodynamic shapes.
• Limitations: the inside/outside of the object is unknown; no volume or mass properties; surfaces may have gaps or overlaps.
• Applications: car bodies, aircraft skins, ship hulls, consumer product casings, dies and moulds.

Solid modelling

The object is represented as a closed volume with a definite inside and outside. Two main schemes:

• CSG (constructive solid geometry): primitives (block, cylinder, sphere, cone) combined with Boolean operations (union, intersection, difference); stored as a binary tree.
• B-rep (boundary representation): faces, edges, vertices with topology.
• Advantages: unambiguous and complete, volume, mass, centre of gravity and moment of inertia available, interference checking, FEA meshing, CAM links.
• Limitations: more memory and computing; free-form shapes are harder.