Euclidean Geometry — Paper 7 cases · \(D_0^1\) vs \(D_1^1\) · scaling invariants

Hierarchical Calculus in Euclidean Geometry

Euclidean geometry is fundamentally a theory of scaling, similarity, and invariants. Many geometric questions are not about “how many units changed”, but about “how the structure changes under rescaling”. This makes the relative derivative \(D_1^1\) a natural operator: it extracts stable exponents such as \(1,2,3,-1\), which encode geometric dimension, curvature scaling, and numerical convergence behavior.

Author: GOSSA AHMED Format: Paper-style Keywords: Similarity · Scaling · Curvature · Mesh 2025
Abstract Introduction Method Cases Decision Guide References
Notation rule: On this website derivatives are written explicitly as \(D_r^n\). The classical differential derivative is \(D_0^1\). The basic relative derivative is \(D_1^1\).

Abstract

This paper-style application page shows that many Euclidean geometry phenomena are better described by structural, scale-invariant operators rather than absolute slopes. Using seven numeric case studies, we compare \(D_0^1\) (absolute change) to \(D_1^1\) (relative/log-derivative). Results demonstrate that \(D_1^1\) acts as an “invariant extractor”: it returns scaling exponents directly, independent of units or current size. These exponents coincide with geometric dimensions, curvature scaling laws, and convergence orders in numerical geometry.

Keywords: Euclidean geometry; similarity; scaling laws; invariants; curvature; numerical mesh; hierarchical calculus.

1) Introduction

In Euclidean geometry, similarity implies that shape properties are preserved under scaling. If an object is rescaled by a factor \(s\), lengths scale as \(s^1\), areas as \(s^2\), and volumes as \(s^3\). These are not numerical coincidences but structural laws.

The classical derivative \(D_0^1\) measures unit-dependent slopes, which are useful for sensitivity analysis but often fail to reveal scale invariance. The relative derivative \(D_1^1\), however, is designed exactly for extracting stable exponents in power-law regimes.

2) Method

We use a scaling parameter \(s>0\) and compare:

\[ D_0^1 y(s)=\frac{dy}{ds}, \qquad D_1^1 y(s)=\frac{d\ln y}{d\ln s}. \]
Central rule: If \(y(s)=C s^a\) with \(C>0\), then: \[ D_1^1 y(s)=a. \] Thus \(D_1^1\) directly returns the scaling exponent.
Operator Meaning Scaling behavior
\(D_0^1\) Absolute change per unit Depends on units and current scale
\(D_1^1\) Relative structural change Constant for power laws (scale-invariant)

3) Case Studies (1–7)

Reading tip: Each case includes (i) the numeric setup, (ii) \(D_0^1\) vs \(D_1^1\), (iii) an additional equation, and (iv) a discussion paragraph.

Case 1 — Scaling a length (CAD / engineering drawing)

A CAD model is rescaled by factor \(s\). Let a segment length be \(L(s)=10s\) cm. Evaluate the two derivatives and interpret their geometric meaning.

\[ L(s)=10s,\quad L(2)=20. \]
Step \(D_0^1\) \(D_1^1\)
Compute \(D_0^1 L=10\) \(D_1^1 L=1\)
Meaning “+10 cm per +1 scale unit” “Length scales like \(s^1\)”

Additional equation (similarity ratio):

\[ \frac{L(s_2)}{L(s_1)}=\frac{s_2}{s_1}. \]
Discussion: The value \(D_1^1=1\) is not a local slope—it is the structural exponent defining the similarity class. Unlike \(D_0^1\), this value is independent of units and remains stable across scales.

Case 2 — Scaling an area (floor plan / tiling)

A square is scaled by \(s\). If \(L(s)=3s\) meters, then \(A(s)=L(s)^2=9s^2\).

\[ A(s)=9s^2,\quad A(2)=36. \]
Step \(D_0^1\) \(D_1^1\)
Compute \(D_0^1A=18s \Rightarrow D_0^1A(2)=36\) \(D_1^1A=2\)
Meaning Absolute sensitivity grows with scale Area scales like \(s^2\)

Additional equation (area law):

\[ A \propto L^2. \]
Discussion: When comparing objects with different sizes, absolute slopes are misleading. The exponent \(2\) remains stable, revealing the geometric dimension of area scaling.

Case 3 — Scaling a volume (3D modeling)

Consider a 3D object with \(V(s)=2s^3\). At \(s=1.5\), \(V=6.75\). The geometric meaning is that volume follows cubic scaling.

\[ V(s)=2s^3,\quad V(1.5)=6.75. \]
Step \(D_0^1\) \(D_1^1\)
Compute \(D_0^1V=6s^2 \Rightarrow D_0^1V(1.5)=13.5\) \(D_1^1V=3\)
Meaning Scale-dependent slope Volume scales like \(s^3\)

Additional equation (volume similarity):

\[ \frac{V_2}{V_1}=\left(\frac{s_2}{s_1}\right)^3. \]
Discussion: The constant exponent \(3\) is a geometric invariant of Euclidean space. \(D_1^1\) reveals this directly, while \(D_0^1\) changes with the current scale.

Case 4 — Similar triangles (ratio invariance)

Let a side be \(a(s)=5s\). The ratio \(a/s=5\) is constant, but \(D_1^1\) detects similarity without constructing ratios manually.

\[ a(s)=5s,\quad \frac{a}{s}=5. \]
Comparison \(D_0^1\) \(D_1^1\)
Similarity detection \(D_0^1(a/s)=0\) (after normalization) \(D_1^1a=1\) directly
Meaning Needs explicit ratio building Normalization is intrinsic

Additional equation (triangle similarity):

\[ \frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}=s. \]
Discussion: Similarity is fundamentally exponent-based. \(D_1^1\) extracts the exponent immediately, making it a clean descriptor for similarity classes.

Case 5 — Curvature scaling (circle expansion)

A circle radius is scaled as \(R(s)=2s\). Curvature is \(k=1/R=\frac{1}{2}s^{-1}\). Curvature therefore follows a negative scaling exponent.

\[ R(s)=2s,\quad k(s)=\frac{1}{2}s^{-1}. \]
Comparison \(D_0^1\) \(D_1^1\)
Derivative \(D_0^1k=-\frac{1}{2}s^{-2}\) \(D_1^1k=-1\)
Meaning Scale-dependent sensitivity Curvature scales as \(s^{-1}\)

Additional equation (general curvature):

\[ k=\frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}}. \]
Discussion: The exponent \(-1\) classifies curvature behavior under scaling. This is crucial in differential geometry and in shape analysis (e.g., smoothing, design curvature constraints).

Case 6 — Mesh refinement (numerical geometry / FEM)

Let mesh step be \(h\) and numerical error follow \(E(h)=Ch^p\). The exponent \(p\) is the convergence order. Consider \(E(h)=4h^2\).

\[ E(h)=4h^2,\quad E(0.2)=0.16. \]
Comparison \(D_0^1\) \(D_1^1\)
Derivative \(D_0^1E=8h \Rightarrow 1.6\) \(D_1^1E=2\)
Meaning Slope depends on \(h\) Convergence order \(p=2\)

Additional equation (extract exponent from two points):

\[ p=\frac{\ln(E_2/E_1)}{\ln(h_2/h_1)}. \]
Discussion: This case shows \(D_1^1\) is not merely conceptual—it is a standard engineering diagnostic tool. In numerical analysis, the exponent is the primary measure of algorithm quality.

Case 7 — Shape descriptor: perimeter vs area (scale-free)

Shape descriptors often combine perimeter \(P\) and area \(A\). Under scaling, \(P\sim s^1\) while \(A\sim s^2\). Let \(P(s)=12s\), \(A(s)=9s^2\).

\[ P(s)=12s,\quad A(s)=9s^2. \]
Quantity \(D_0^1\) \(D_1^1\)
Perimeter \(D_0^1P=12\) \(D_1^1P=1\)
Area \(D_0^1A=18s\) \(D_1^1A=2\)

Additional equation (scale-invariant descriptor):

\[ Q=\frac{P^2}{A}\quad \Rightarrow\quad Q \text{ is invariant under scaling}. \]
Discussion: Many geometric quality measures are built from combinations like \(P^2/A\) because they remove scale effects. This highlights that structural invariants dominate geometric reasoning.

4) Decision Guide: When to use \(D_0^1\) vs \(D_1^1\)

Question Recommended operator Reason
How much did the value change numerically? \(D_0^1\) Absolute sensitivity / reporting
How does the quantity behave under rescaling? \(D_1^1\) Extracts stable exponents / invariants
Comparing shapes of different sizes \(D_1^1\) Scale-free comparison
Takeaway: If the phenomenon is dominated by similarity/scaling laws, \(D_1^1\) is the stable descriptor.

References

\[ [1]\;\text{Euclid. Elements. Classical foundations of Euclidean geometry.} \] \[ [2]\;\text{Coxeter, H. S. M. (1969). Introduction to Geometry.} \] \[ [3]\;\text{Kreyszig, E. Differential Geometry. (Curvature & invariants).} \] \[ [4]\;\text{Bathe, K. J. Finite Element Procedures. (Error scaling).} \] \[ [5]\;\text{GOSSA AHMED (2025). Hierarchical Calculus (Official Website).} \]

Citation

\[ \texttt{GOSSA\ AHMED.\ Hierarchical\ Calculus\ in\ Euclidean\ Geometry:\ Paper-Style\ with\ 7\ Real\ Examples\ (D_0^1\ vs\ D_1^1).\ Official\ Website.\ (2025).} \]