# 28. Geometric design: representations, equivalence, and conditioning

Research note, dive 14, 2026-09-08. Mathematical groundwork independent of a particular application.

## Primary reading

Thomas W. Sederberg's *Computer Aided Geometric Design* develops curves through blending
functions and geometric constructions. The archived copy has a 2012 repository cover and
**July 25, 2017 course-note footers**. Selected sections establish Bernstein representations,
subdivision, degree elevation, derivatives, and homogeneous rational curves. Read physical PDF
pages 34–37, 39–40, 42–44, 65; printed pages 18–21, 23–24, 26–28, 49.
[SRC: sederberg-cagd] [Local source and reading record](../library/notes/sederberg-cagd.md).

## 1. Derivation: partition of unity does real work

Let control points P_i be vectors in R^d, and t a dimensionless parameter in [0,1]. Define

\[
B_i^n(t)={n\choose i}t^i(1-t)^{n-i},\qquad c(t)=\sum_{i=0}^n B_i^n(t)P_i.
\]

The binomial theorem gives sum_i B_i^n(t)=1. Nonnegativity on this interval makes c(t) a convex
combination. Thus any convex region containing all control points also contains the curve.
This is a sufficient containment certificate; it is not necessary that a tight curve enclosure
contain every control point. Outside [0,1], the positivity argument no longer applies.

For an affine transformation T(x)=Ax+b, direct substitution gives

\[
\sum_i B_i^n(t)T(P_i)=A c(t)+b\sum_i B_i^n(t)=T(c(t)).
\]

This is **equivariance** under transforming the geometry, often called affine invariance of the
construction. It does not say the curve's coordinates, length, or curvature remain unchanged.
A nonuniform affine scale changes Euclidean lengths and angles.

## 2. Derivation: exact representation changes need not change shape

Use the identities

\[
(1-t)B_i^n=\frac{n+1-i}{n+1}B_i^{n+1},\qquad
tB_i^n=\frac{i+1}{n+1}B_{i+1}^{n+1}.
\]

Multiplication by (1-t)+t=1 and collection of coefficients gives a degree n+1 representation:

\[
Q_0=P_0,\quad Q_{n+1}=P_n,\quad
Q_i=\frac{i}{n+1}P_{i-1}+\left(1-\frac{i}{n+1}\right)P_i\quad(1\le i\le n).
\]

The function c(t) is identical. A quadratic with points (0,0),(1,2),(2,0) becomes a cubic with
points (0,0),(2/3,4/3),(4/3,4/3),(2,0). The parameter array has changed length while the represented
geometry has not. A distance between raw control arrays therefore needs compatible representations
or an explicit correspondence. Degree elevation alone is not additional geometric detail.

**Source correction, locally verified:** the third equality on printed page 23, equation (2.7),
contains `(1+t)B_i^n(t)` where `(1-t)B_i^n(t)` is required. Visual inspection of the PDF confirms
this is printed, not text-extraction corruption. The final degree-elevation formula (2.8) is
consistent with the derivation above. The original PDF is preserved unchanged.

## 3. Derivation: smooth joins depend on parameter duration

Differentiating the basis and collecting adjacent terms yields, for normalized t,

\[
c'(t)=n\sum_{i=0}^{n-1}B_i^{n-1}(t)(P_{i+1}-P_i).
\]

For physical parameter s in an interval of length h>0, derivatives acquire a factor 1/h.
Two segments of degrees n,m and durations h_1,h_2 join with C^1 continuity when

\[
P_n=Q_0,\qquad \frac n{h_1}(P_n-P_{n-1})=\frac m{h_2}(Q_1-Q_0).
\]

Matching only positively aligned, nonzero tangent directions establishes an oriented geometric
G^1 join; matching speed establishes C^1 for the declared parameter. Changing timing can break
C^1 without changing the curve's trace. At zero tangent, regular-curve reasoning needs a separate
analysis. In particular, a zero derivative alone does not prove a geometric cusp:
c(t)=(t^3,0) has zero derivative at zero and a straight trace. This qualifies the unqualified
hodograph/cusp sentence on printed page 27.

## 4. Derivation: rational weights introduce a gauge freedom

For weights w_i and D(t)=sum_i w_i B_i^n(t),

\[
c_w(t)=\frac{\sum_i w_i B_i^n(t)P_i}{D(t)},\qquad D(t)\ne0.
\]

Strictly positive weights ensure D>0 on [0,1]. Multiplying every weight by the same nonzero
constant leaves c_w unchanged. Thus the representation has at least one redundant scale direction.
Differentiating with respect to a weight gives

\[
\frac{\partial c_w}{\partial w_j}=\frac{B_j^n(t)}{D(t)}(P_j-c_w(t)).
\]

Small denominators can amplify sensitivity. Weight positivity, normalization, and denominator
bounds are different conditions; none establishes a good perceptual metric. Homogeneous coordinates
(w_i P_i,w_i) permit polynomial constructions before projection, but derivatives after projection
require the quotient rule.

## Boundaries and continuation

These derivations concern geometry, not raster visibility, self-intersection avoidance, topology,
or the semantics of an image. Next readings should develop B-spline knot insertion and local
support, interval-certified intersection, and curvature regularization. Subdivision, alternative
bases, and rational gauges belong in any comparison of parameter-space distances.
Executable illustrations are in [the mathematical checks](../experiments/05_research_math.py).



Further reading, dive 15: [33](33_spline_spaces.md) now develops the B-spline and knot-insertion branch, including sampling loss and refinement-aware penalties.