Representation of Surfaces and Solids

CAD surface entities are divided into analytic and synthetic surfaces.

Analytic surfaces

• Plane surface: defined by three non-collinear points, or a point and a normal. Equation $ax+by+cz=d$.
• Ruled (lofted) surface: generated by joining corresponding points of two boundary curves with straight lines: $\mathbf P(u,w)=(1-w)\mathbf P_1(u)+w\mathbf P_2(u)$. Example: a cone or a twisted turbine-blade strip.
• Tabulated cylinder: a planar curve (directrix) translated along a straight line of fixed direction and length: $\mathbf P(u,w)=\mathbf C(u)+w\mathbf L$. Example: an extruded profile.
• Surface of revolution: a plane curve (generatrix) rotated through an angle about an axis: $\mathbf P(u,\phi)$ with the radius from the curve. Examples: bottles, vases, pulleys, domes.

Synthetic surfaces

These are free-form surfaces for shapes with no simple equation: car bodies, aircraft wings, ship hulls, helmets and plastic housings. They are built from a net of control points (Bezier, B-spline) or from four boundary curves (Coons patch).

Surfaces are stored in parametric form $\mathbf P(u,w)$, $0\le u,w\le1$, so normals, tangents and boundaries are easy to compute for display and machining.

Equation of plane

Two vectors in the plane:

$$\mathbf{AB}=(-1,2,0),\qquad \mathbf{AC}=(-1,0,3)$$

Normal:

$$\mathbf n=\mathbf{AB}\times\mathbf{AC}= \begin{vmatrix}\mathbf i&\mathbf j&\mathbf k\\-1&2&0\\-1&0&3\end{vmatrix} =(6-0)\mathbf i-(-3-0)\mathbf j+(0+2)\mathbf k=(6,\ 3,\ 2)$$

Plane: $6x+3y+2z=d$. Using A: $d=6$.

$$6x+3y+2z=6\quad\Longleftrightarrow\quad \frac{x}{1}+\frac{y}{2}+\frac{z}{3}=1$$

Check B: $6(0)+3(2)+0=6$. Check C: $0+0+2(3)=6$.

Unit normal

$|\mathbf n|=\sqrt{36+9+4}=7$, so $\hat{\mathbf n}=(0.857,\ 0.429,\ 0.286)$.

Parametric form

$$\mathbf P(u,w)=\mathbf A+u\,\mathbf{AB}+w\,\mathbf{AC}=(1-u-w,\ 2u,\ 3w)$$

where $u,w$ in $[0,1]$ give the part of the plane bounded by the parallelogram on $\mathbf{AB}$ and $\mathbf{AC}$.

Distance of Q(2, 2, 2)

$$D=\frac{|6(2)+3(2)+2(2)-6|}{7}=\frac{16}{7}=2.286$$

Answer: plane 6x + 3y + 2z = 6; unit normal (0.857, 0.429, 0.286); distance of Q = 2.286 units.

A ruled surface is generated by moving a straight line (a ruling) so that its end points slide along two boundary curves. Corresponding points of the two curves (same $u$) are joined by straight lines:

$$\mathbf P(u,w)=(1-w)\,\mathbf P_1(u)+w\,\mathbf P_2(u),\qquad 0\le u,w\le1$$

The surface is linear in $w$; it is a plane if the two curves are coplanar and straight, otherwise a twisted (hyperbolic-paraboloid type) patch.

Boundary curves

$$\begin{aligned} \mathbf P_1(u)&=(0,0,0)+u(6,0,0)=(6u,\ 0,\ 0) \\ \mathbf P_2(u)&=(0,4,2)+u(6,2,3)=(6u,\ 4+2u,\ 2+3u) \end{aligned}$$

Surface equation

$$\mathbf P(u,w)=\big(6u,\ w(4+2u),\ w(2+3u)\big)$$

Points

For example, at $(0.5,0.75)$: $y=0.75\times5=3.75$, $z=0.75\times3.5=2.625$.

Answer: P(u,w) = (6u, w(4+2u), w(2+3u)); points (3, 2.5, 1.75), (1.5, 2.25, 1.375), (3, 3.75, 2.625).

A surface of revolution is formed by rotating a plane curve (the generatrix or profile) about an axis in its plane. Every point of the curve moves on a circle (parallel); each position of the curve is a meridian. For a profile $r=f(s)$, $z=g(s)$ about the Z axis:

$$\mathbf P(s,\phi)=\big(f(s)\cos\phi,\ f(s)\sin\phi,\ g(s)\big),\qquad 0\le\phi\le2\pi$$

Generatrix

The line is $r=2+2s$, $z=6s$ for $0\le s\le1$.

$$\mathbf P(s,\phi)=\big((2+2s)\cos\phi,\ (2+2s)\sin\phi,\ 6s\big)$$

The surface is the side of a frustum of a cone with bottom radius 2, top radius 4 and height 6.

Points

Curved surface area

Slant length $l=\sqrt{(4-2)^2+6^2}=\sqrt{40}=6.325$.

$$A=\pi\,(r_1+r_2)\,l=\pi(2+4)(6.325)=119.2\ \text{square units}$$

Answer: P(s, phi) = ((2+2s)cos phi, (2+2s)sin phi, 6s); points (0,3,3), (-4,0,6), (1.768,1.768,1.5); area = 119.2 square units.

A tabulated cylinder is a surface generated by translating a plane curve (the directrix) along a straight line of fixed direction and finite length (the generator). All generators are parallel. It is the surface produced by an extrude feature in solid modelling; if the directrix is a circle and the direction is perpendicular to its plane, a right circular cylinder results.

Equation

$$\mathbf P(u,w)=\mathbf C(u)+w\,\mathbf L,\qquad 0\le u\le1\ (\text{curve parameter}),\ 0\le w\le1$$

where $\mathbf C(u)$ is the directrix and $\mathbf L$ is the vector giving the direction and length of translation.

Numerical

$$\mathbf P(u,w)=\big(10\cos u,\ 10\sin u,\ 25w\big)$$

At $u=30^\circ$ and $w=0.4$:

$$\begin{aligned} x&=10\cos30^\circ=8.660 \\ y&=10\sin30^\circ=5.000 \\ z&=25(0.4)=10.0 \end{aligned}$$

The surface is a quarter of a cylinder of radius 10 and height 25, with its curved area $=\tfrac14(2\pi\times10\times25)=392.7$ square units.

Answer: P(u,w) = C(u) + wL; the point is (8.660, 5.000, 10.0).

A Bezier surface is a parametric surface patch defined by a rectangular net (mesh) of control points. It is the tensor product of two Bezier curves, one in the $u$ direction and one in the $w$ direction.

Equation

For an $(m+1)\times(n+1)$ net of control points $\mathbf B_{ij}$:

$$\mathbf P(u,w)=\sum_{i=0}^{m}\sum_{j=0}^{n}\mathbf B_{ij}\,J_{m,i}(u)\,J_{n,j}(w),\qquad 0\le u,w\le1$$

where $J_{m,i}(u)=\binom mi u^i(1-u)^{m-i}$ is the Bernstein basis. The bicubic patch has $m=n=3$, so 16 control points:

$$\mathbf P(u,w)=\sum_{i=0}^{3}\sum_{j=0}^{3}\mathbf B_{ij}J_{3,i}(u)J_{3,j}(w)$$

Properties

1. The degree in $u$ is $m$ and in $w$ is $n$ (one less than the number of points in that direction).
2. The four corner points of the net lie on the surface: $\mathbf P(0,0)=\mathbf B_{00}$, $\mathbf P(1,0)=\mathbf B_{m0}$, etc.
3. The four boundary curves are Bezier curves defined by the boundary rows and columns of the net.
4. Tangent planes at the corners are spanned by the adjacent net edges.
5. Convex hull: the surface lies in the convex hull of the control net.
6. Global control: moving any control point changes the whole patch.
7. Invariant under affine transformation (transform the net only).
8. Two patches join with $C^0$ continuity if the edge rows coincide; with $C^1$ if the adjacent rows are collinear with equal ratio.

Difference from curve

 Curve:   1 parameter u,   row of points B_i
 Surface: 2 parameters u,w, net of points B_ij

   B03---B13---B23---B33
    |     |     |     |      u ->
   B02---B12---B22---B32
    |     |     |     |      w
   B01---B11---B21---B31      down
    |     |     |     |
   B00---B10---B20---B30

Only the corners lie on the surface; the interior points only pull it.

Equation of the patch

For degree 2 in both directions:

$$\mathbf P(u,w)=\sum_{i=0}^{2}\sum_{j=0}^{2}\mathbf B_{ij}\,J_{2,i}(u)\,J_{2,j}(w)$$

with $J_{2,0}=(1-t)^2$, $J_{2,1}=2t(1-t)$, $J_{2,2}=t^2$.

Control net

Point at (0.5, 0.5)

Weights for $t=0.5$ in both directions: $0.25,\ 0.5,\ 0.25$.

$x$ and $y$ follow from the grid ($x=4i$, $y=4j$ at the net points):

$$x=4\,(0\cdot0.25+1\cdot0.5+2\cdot0.25)=4,\qquad y=4$$

For $z$, first sum across $j$ for each row ($w$ weights $0.25, 0.5, 0.25$):

$$\begin{aligned} \text{row 0: }&0(0.25)+2(0.5)+0(0.25)=1.0 \\ \text{row 1: }&3(0.25)+6(0.5)+3(0.25)=4.5 \\ \text{row 2: }&1.0 \end{aligned}$$

Then across $i$ ($u$ weights $0.25, 0.5, 0.25$):

$$z=0.25(1.0)+0.5(4.5)+0.25(1.0)=2.75$$

Surface point $(4,\ 4,\ 2.75)$ (the central net point has $z=6$, so the surface lies well below it, as expected for an approximating surface).

Point at (0.25, 0.5)

$u$ weights for $t=0.25$: $(0.75)^2=0.5625$, $2(0.25)(0.75)=0.375$, $(0.25)^2=0.0625$. The $w$ weights remain $0.25, 0.5, 0.25$, so the row sums are again $1.0, 4.5, 1.0$.

$$\begin{aligned} x&=4(0.375+2\times0.0625)=2.0 \\ y&=4(0.5+2\times0.25)=4.0 \\ z&=0.5625(1.0)+0.375(4.5)+0.0625(1.0)=2.3125 \end{aligned}$$

Answer: P(0.5, 0.5) = (4, 4, 2.75); P(0.25, 0.5) = (2, 4, 2.3125).

A B-spline surface is a tensor-product surface built from B-spline basis functions in two directions. It is defined by a control net and two knot vectors.

Equation

For an $(m+1)\times(n+1)$ net $\mathbf B_{ij}$, with orders $k$ in $u$ and $l$ in $w$:

$$\mathbf P(u,w)=\sum_{i=0}^{m}\sum_{j=0}^{n}\mathbf B_{ij}\,N_{i,k}(u)\,M_{j,l}(w)$$

$N_{i,k}$ and $M_{j,l}$ are B-spline basis functions defined by the knot vectors $U$ and $W$ (Cox-de Boor recursion). A bicubic B-spline surface uses $k=l=4$.

Comparison

Advantages

1. A large, complex surface (car roof, wing) is one entity with a low degree.
2. Local modification does not disturb the rest of the surface.
3. Smooth ($C^2$ for bicubic) without extra constraints.
4. Convex hull property and affine invariance hold.
5. The rational version (NURBS) also represents exact cylinders, cones, spheres and tori, and is the standard in CAD and IGES/STEP exchange.

A Coons patch is a surface patch defined by its four boundary curves (not by a net of control points). The interior is obtained by blending the boundaries, so the patch exactly passes through all four edges. It is used to fill a four-sided region bounded by given curves, such as a gap between existing surfaces.

Bilinearly blended Coons surface

$$\begin{aligned} \mathbf P(u,w)={}&(1-w)\mathbf P(u,0)+w\,\mathbf P(u,1)+(1-u)\mathbf P(0,w)+u\,\mathbf P(1,w) \\ &-\big[(1-u)(1-w)\mathbf P_{00}+u(1-w)\mathbf P_{10}+(1-u)w\,\mathbf P_{01}+uw\,\mathbf P_{11}\big] \end{aligned}$$

The first four terms are two ruled surfaces; the bracket (the bilinear surface through the corners) is subtracted because the corners are counted twice.

Corner compatibility

$\mathbf P_{00}=(0,0,0)$, $\mathbf P_{10}=(4,0,0)$, $\mathbf P_{01}=(0,3,0)$, $\mathbf P_{11}=(4,3,2)$.

The curves agree at every corner (e.g. $P(u{=}1,w{=}1)=(4,3,2)$ from both the top and right curves), so the patch is valid.

At (0.5, 0.5)

Boundary points: $\mathbf P(0.5,0)=(2,0,0)$, $\mathbf P(0.5,1)=(2,3,0.5)$, $\mathbf P(0,0.5)=(0,1.5,0)$, $\mathbf P(1,0.5)=(4,1.5,0.5)$.

Ruled terms: $0.5(2,0,0)+0.5(2,3,0.5)+0.5(0,1.5,0)+0.5(4,1.5,0.5)=(4,3,0.5)$

Corner term: $0.25[(0,0,0)+(4,0,0)+(0,3,0)+(4,3,2)]=0.25(8,6,2)=(2,1.5,0.5)$

$$\mathbf P(0.5,0.5)=(4,3,0.5)-(2,1.5,0.5)=(2,\ 1.5,\ 0)$$

At (0.5, 0.25)

Boundary points: $\mathbf P(0.5,0)=(2,0,0)$, $\mathbf P(0.5,1)=(2,3,0.5)$, $\mathbf P(0,0.25)=(0,0.75,0)$, $\mathbf P(1,0.25)=(4,0.75,0.125)$.

Ruled terms: $0.75(2,0,0)+0.25(2,3,0.5)+0.5(0,0.75,0)+0.5(4,0.75,0.125)=(1.5+0.5+0+2,\ 0.75+0.375+0.375,\ 0.125+0.0625)=(4,\ 1.5,\ 0.1875)$

Corner term: $(0.5)(0.75)(0,0,0)+(0.5)(0.75)(4,0,0)+(0.5)(0.25)(0,3,0)+(0.5)(0.25)(4,3,2)=(1.5+0.5,\ 0.375+0.375,\ 0.25)=(2,\ 0.75,\ 0.25)$

$$\mathbf P(0.5,0.25)=(4,1.5,0.1875)-(2,0.75,0.25)=(2,\ 0.75,\ -0.0625)$$

Answer: P(0.5, 0.5) = (2, 1.5, 0); P(0.5, 0.25) = (2, 0.75, -0.0625).

(a) Fillet surface

A fillet surface is a smooth blending surface that joins two intersecting surfaces (or a surface and a plane) with a rounded transition. It removes sharp edges.

• Constant-radius fillet: the surface is the envelope of a ball of radius $r$ rolling in contact with both surfaces; the centre follows the line at distance $r$ from both. The section is a circular arc tangent to both surfaces.
• Variable-radius fillet: the radius changes along the edge.
• A chamfer is the flat version.
• If the two faces meet at an angle, the fillet arc has the same radius but a different arc length.

   before            after fillet
   |                 |
   |____             |__
                        `--.____   (rounded corner)

Uses: reduces stress concentration in machine parts, makes castings and moulds easier to manufacture (draft and flow), improves appearance and safety, and avoids sharp edges in sheet metal and plastic parts.

(b) Offset surface

An offset surface is a surface at a constant distance $d$ along the normal from a given surface:

$$\mathbf P_o(u,w)=\mathbf P(u,w)+d\,\hat{\mathbf n}(u,w)$$

Examples: offsetting a cylinder of radius 20 by $d=3$ gives radius 23 (outward) or 17 (inward); a plane offset is a parallel plane.

• Limits: if $d$ is larger than the smallest radius of curvature on the concave side, the offset self-intersects (a cusp or loop appears).
• Uses: cutter-path generation (offset the part surface by the tool radius so the tool centre follows it), shelling and wall thickness (hollow parts, plastic housings), clearance and tolerance zones, mould cavity from a part with shrinkage allowance, and sheet-metal thickness.