> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # Projection Possibility Space Series ## Paper 02 — Dynamic Projection Geometry ### Animated Interaction as a Continuous Geometric Experiment: From 3D Motion and 2.5D Intermediate States to the Closure of 2D Projection Events ### Animation Interaction as a Continuous Geometric Experiment: From 3D Motion and 2.5D Intermediate States to 2D Projection-Event Closure **Document ID:** `PPS-DPG-02-v0.1` **Version:** v0.1 **Date:** 2026-09-21 **Author:** Neo.K **AI Research Collaborator:** Aletheia / ChatGPT, GPT-5.6 Sol **Institution:** EveMissLab / 一言諾科技有限公司 **Series:** Projection Possibility Space / Projection Possibility Space Series **Prerequisite:** Paper 01 — 3D Projection Possibility Space **Status:** `FOUNDATIONAL THEORY / INTERACTIVE GEOMETRIC EXPERIMENT FRAMEWORK` --- # Abstract Planar geometry problems are usually studied directly through two-dimensional coordinates, boundary curves, support functions, finite parameter families or discrete figures. Yet a family of two-dimensional configurations need not be generated most naturally by two dimensions themselves. For many geometric problems, three-dimensional objects, relative poses, sections, occlusion and projection directions can naturally generate a continuous family of two-dimensional configurations. This paper proposes: $$ \boxed{ \textbf{Dynamic Projection Geometry} } $$ Its core is not to draw a two-dimensional problem as a three-dimensional picture, but to treat three-dimensional space as a generator of the two-dimensional possibility space. Let the three-dimensional source body be: $$ M, $$ the time-dependent transformation be: $$ T_t, $$ and the projection operator be: $$ \Pi_t. $$ The dynamic family of two-dimensional geometries is then written as: $$ \boxed{ G_t = \Pi_t(T_tM). } $$ Animation is therefore not merely visualization, but a continuous path through configuration space: $$ \gamma: t \mapsto G_t. $$ The genuinely important objects of study are not just a sequence of frames, but: 1. which projected geometric types occur along the path; 2. which critical events change combinatorial / contact / visibility topology; 3. which geometric states exist only in extremely short transition windows; 4. whether finite static sampling can miss these events; 5. how to upgrade animated interaction into continuous geometric experiments that are auditable, replayable and able to output counterexample certificates. The paper further defines: $$ 3D \rightarrow 2.5D \rightarrow 2D $$ a three-layer projection architecture. Here 2.5D is not the name of a visual style, but an intermediate geometric state retaining depth, layer order, occlusion, visibility provenance and local projection sources. This allows the final two-dimensional result to retain enough information to trace: $$ \boxed{ \text{which 3D structure and which motion generated a particular 2D boundary event.} } $$ Finally, the paper proposes the Dynamic Projection Certificate, Projection Event Ledger, Adaptive Critical-Event Sampling and Interactive Counterexample Loop, allowing animated interaction to advance from a presentation layer into a component of geometric closure and stress testing. --- # Keywords dynamic projection geometry、3D-to-2D projection、2.5D state、interactive geometry、continuous geometric experiment、projection event、critical frame、configuration space、counterexample search、geometric closure、animation interaction、visibility topology --- # 0. Research Position This paper does not study ordinary computer animation. Nor does it study: $$ \boxed{ \text{making mathematical figures into attractive 3D animations.} } $$ The actual question is: > If a two-dimensional geometric configuration can be generated by a three-dimensional space with more degrees of freedom, should we construct the source body directly in three dimensions and let three-dimensional motion continuously unfold two-dimensional possibilities, rather than manually enumerate only finitely many planar cases? The paper answers: $$ \boxed{ \text{Yes; animation itself can also become a geometric search operator.} } $$ Therefore: $$ \text{Animation} \neq \text{Presentation only}. $$ A more appropriate interpretation is: $$ \boxed{ \text{Animation} = \text{continuous traversal of a geometric state space}. } $$ --- # 1. The Three-Dimensional Source Body Is a Generator, Not the Result Let: $$ M\subset\mathbb R^3 $$ be a three-dimensional source body. It can be: - rigid body； - convex body； - polyhedral body； - smooth surface； - articulated system； - curve / tube； - deformable mesh； - implicit field； - multiple-body assembly。 If we merely hold $M$ fixed and project it once, we obtain: $$ G = \Pi(M). $$ This is only a single two-dimensional result. Dynamic Projection Geometry concerns a family of transformations: $$ T_t, \qquad t\in I, $$ and a projection that may also vary with time: $$ \Pi_t. $$ Therefore: $$ \boxed{ G_t = \Pi_t(T_tM). } $$ The entire object of study is: $$ \boxed{ \mathcal G_\gamma = \{G_t:t\in I\}. } $$ where: $$ \gamma $$ denotes this dynamic projection path. --- # 2. The Configuration-Space View Let the configuration space of the three-dimensional system be: $$ \mathcal C_{3D}. $$ If $M$ is rigid, its most basic component can include: $$ SE(3). $$ If deformation parameters are also included: $$ \lambda = (\lambda_1,\ldots,\lambda_k), $$ then: $$ \mathcal C_{3D} \supset SE(3)\times\Lambda. $$ Projection settings form a separate space: $$ \mathcal P. $$ The complete generating space can therefore be written as: $$ \boxed{ \mathcal C = \mathcal C_{3D} \times \mathcal P. } $$ Animation is a path: $$ \boxed{ \Gamma: I \rightarrow \mathcal C. } $$ and the two-dimensional geometry is given by the projection map: $$ \mathscr P: \mathcal C \rightarrow \mathcal G_{2D}. $$ Thus: $$ \boxed{ G_t = \mathscr P(\Gamma(t)). } $$ This formalization reinterprets animation as: $$ \boxed{ \text{a path through the preimage space of 2D geometries}. } $$ --- # 3. Why 3D Can Unfold 2D Possibilities If we construct directly in the plane: $$ G(\alpha_1,\ldots,\alpha_n), $$ we usually need to know in advance: - which parameters matter; - which branches exist; - which contact topologies may occur; - which degenerate states are worth enumerating. A three-dimensional source body, however, can generate a family of projections through natural motion: $$ \Pi(T_tM). $$ Some planar phenomena in this family need not be explicitly encoded in advance. For example: - a projected edge appears; - a projected edge disappears; - two projected vertices merge； - apparent cusp formation； - occlusion swap； - support direction switching； - silhouette topology change； - contact ordering reversal。 Thus three dimensions do not add a visual dimension. They add: $$ \boxed{ \text{degrees of freedom for generating two-dimensional states.} } $$ --- # 4. The Fundamental Difference Between Animation and Static Enumeration Static enumeration usually produces: $$ G_{t_1}, G_{t_2}, \ldots, G_{t_N}. $$ Animation research instead concerns: $$ \{G_t\}_{t\in I}. $$ These are different. There may exist: $$ t^\star \in (t_i,t_{i+1}) $$ such that: $$ G_{t^\star} $$ has an entirely different critical structure, while neither endpoint: $$ G_{t_i}, \qquad G_{t_{i+1}} $$ reveals it. Therefore: $$ \boxed{ \text{finite frame sampling} \not\Rightarrow \text{continuous event coverage}. } $$ --- # 5. Projection Event Define: $$ E $$ to be a projection event if a projected structural quantity undergoes, at: $$ t=t^\star $$ a qualitative change. Examples: $$ \boxed{ E_{\rm contact} } $$ the number of contacts changes; $$ \boxed{ E_{\rm visibility} } $$ visibility ordering changes; $$ \boxed{ E_{\rm support} } $$ the support winner changes; $$ \boxed{ E_{\rm topology} } $$ the projected boundary topology changes; $$ \boxed{ E_{\rm degeneracy} } $$ degenerate contact, a repeated root or collinearity occurs; $$ \boxed{ E_{\rm branch} } $$ the projected configuration switches from one branch to another. --- # 6. Event Functions An event can usually be described by: $$ f_j(t) $$ . A critical event occurs at: $$ \boxed{ f_j(t^\star)=0. } $$ For example: ## Contact $$ f_{\rm c}(t) = d(A_t,B_t). $$ If: $$ f_{\rm c}(t^\star)=0, $$ there is contact. ## Support switch $$ f_{\rm s}(t) = h_A(\theta,t) - h_B(\theta,t). $$ If: $$ f_{\rm s}(t^\star)=0, $$ a support-winner exchange may occur. ## Visibility swap If two surface points project in the same pixel direction and their depth difference is: $$ f_{\rm z}(t) = z_1(t)-z_2(t), $$ then: $$ f_{\rm z}(t^\star)=0 $$ is a visibility-order event. --- # 7. A Critical Frame Is Not an Ordinary Frame An ordinary animation frame: $$ F_k $$ is merely a sample in time. A critical frame, by contrast, is defined by an event condition: $$ \boxed{ F^\star = G_{t^\star}, \qquad f_j(t^\star)=0. } $$ A critical frame should therefore be generated by: $$ \text{event isolation} $$ rather than simply by: $$ \text{FPS sampling}. $$ --- # 8. Finite Sampling Can Miss Events Permanently Consider: $$ f(t) = (t-a)(t-b), $$ where: $$ t_i0, $$ and: $$ f(t_{i+1})>0, $$ we may miss: $$ f(t)<0 $$ throughout the intervening interval. Thus: $$ \boxed{ \text{same endpoint classification} \not\Rightarrow \text{same classification throughout the interval}. } $$ This is why animated interaction must be accompanied by event-aware sampling. --- # 9. The Three State Layers of Dynamic Projection This paper divides projection into: $$ \boxed{ 3D \rightarrow 2.5D \rightarrow 2D. } $$ This is not a ranking of rendering quality. It is a hierarchy of information retention. --- # 10. 3D Source State Denote the three-dimensional source state by: $$ X_t^{(3)}. $$ It can contain at least: $$ \boxed{ X_t^{(3)} = ( M_t, T_t, C_t, L_t ) } $$ where: - $M_t$: geometric source body; - $T_t$：transform state； - $C_t$：camera / projection state； - $L_t$：layer / object provenance。 The three-dimensional layer retains the complete generating information. --- # 11. 2.5D Intermediate State Define: $$ X_t^{(2.5)}. $$ It retains planar position: $$ (x,y), $$ but does not immediately discard: - depth； - layer identity； - source primitive； - occlusion ordering； - visible / hidden flag； - local normal； - source-body label。 It can therefore be written as: $$ \boxed{ X_t^{(2.5)} = \{ (x_i,y_i,d_i,\ell_i,v_i,p_i) \}_i. } $$ where: - $d_i$：depth； - $\ell_i$：layer； - $v_i$：visibility； - $p_i$：provenance。 --- # 12. Why Not Go Directly from 3D to 2D? If we go directly: $$ 3D \rightarrow 2D, $$ we may obtain the same planar outline but lose: $$ \boxed{ \text{which 3D configuration produced it}. } $$ For example, two entirely different three-dimensional configurations: $$ X_A^{(3)} \neq X_B^{(3)} $$ may satisfy: $$ \Pi(X_A^{(3)}) = \Pi(X_B^{(3)}). $$ If only the final 2D result is retained: $$ G, $$ their generation histories cannot be distinguished. The 2.5D layer therefore retains: $$ \boxed{ \text{projection provenance}. } $$ --- # 13. Projection Fiber For a two-dimensional configuration: $$ G, $$ define its projection fiber: $$ \boxed{ \mathcal F_G = \{ X\in\mathcal C: \mathscr P(X)=G \}. } $$ The same: $$ G $$ may have many different three-dimensional preimages. Dynamic Projection Geometry therefore studies not only: $$ G. $$ but also: $$ \boxed{ \mathcal F_G. } $$ This distinguishes identical two-dimensional results by their different generating causes. --- # 14. Animation Actually Sweeps a Neighbourhood of the Projection Fiber If: $$ G_t $$ remains nearly unchanged over a short time, this does not imply that the three-dimensional state has not changed substantially. It is possible that: $$ \Gamma(t) $$ moves along directions in: $$ \mathcal F_G $$ . At some critical time: $$ t^\star, $$ it leaves the fiber-stable region. The two-dimensional result then suddenly undergoes: $$ \boxed{ \text{projection bifurcation}. } $$ --- # 15. Projection-Stratum Stratify configuration space: $$ \mathcal C $$ by projected combinatorial type: $$ \boxed{ \mathcal C = \bigcup_\alpha \mathcal S_\alpha. } $$ Within one stratum: $$ \mathcal S_\alpha, $$ the projection retains the same basic topology / winner ordering / visibility ordering. The stratum boundary: $$ \partial\mathcal S_\alpha $$ contains projection events. --- # 16. Animation Is a Path Across Strata Animation: $$ \Gamma(t) $$ can successively traverse: $$ \mathcal S_{\alpha_1} \rightarrow \mathcal S_{\alpha_2} \rightarrow \cdots \rightarrow \mathcal S_{\alpha_m}. $$ One of animation's actual research outputs is therefore: $$ \boxed{ \text{stratum itinerary}. } $$ For example: ```text S12 → tangency event → S18 → visibility swap → S21 → contact merge → S9 ``` This is closer to a mathematical certificate than merely saving a video. --- # 17. Generic Finite-Event Proposition If: 1. the time interval: $$ I=[a,b] $$ is compact; 2. event functions： $$ f_1,\ldots,f_N $$ are real analytic on each local chart; 3. no: $$ f_j $$ is identically zero on a nonempty time interval; 4. event zeros have no infinite accumulation near chart switches; then event times form a discrete set in each local chart. On a compact interval, if we additionally assume a finite chart cover and no event accumulation, we obtain: $$ \boxed{ \text{finite projection-event decomposition}. } $$ That is, there exist: $$ a=t_0 Given equations, how does the state evolve over time? Dynamic Projection Geometry need not involve physical time at all. Here: $$ t $$ need only be a path parameter. Thus: $$ \boxed{ t \text{ need not equal physical time}. } $$ It can represent: - interpolation； - homotopy； - sweep parameter； - camera angle； - deformation magnitude； - search trajectory。 --- # 46. Relation to Homotopy An animation path: $$ G_t $$ can essentially be regarded as a kind of geometric homotopy. Dynamic Projection Geometry additionally emphasizes: 1. higher-dimensional source provenance; 2. 2.5D intermediate state； 3. projection event detection； 4. interactive path editing； 5. certificate / counterexample output。 Thus: $$ \boxed{ \text{Dynamic Projection Geometry} \supset \text{visual homotopy use case}. } $$ --- # 47. Relation to Motion Planning Motion planning usually studies: $$ \Gamma: [0,1] \rightarrow \mathcal C_{\rm free} $$ to avoid collision. This paper may instead deliberately seek: $$ \partial\mathcal C_{\rm free}. $$ because contact / degeneracy are precisely the high-information regions. Therefore: $$ \boxed{ \text{motion planning avoids critical sets;} } $$ whereas: $$ \boxed{ \text{projection stress testing seeks critical sets.} } $$ --- # 48. Relation to Computational Geometry Computational geometry has extensively studied: - arrangements； - visibility； - convex hull； - Voronoi； - kinetic data structures。 Dynamic Projection Geometry is particularly close to kinetic geometry. The paper's additional emphasis is: $$ \boxed{ \text{higher-dimensional source} \rightarrow \text{interactive projected possibility space} \rightarrow \text{certificate closure}. } $$ It therefore does not replace existing computational geometry, but integrates multiple tools into an interactive projection-search runtime. --- # 49. Kinetic Certificate View We can borrow the ideas of kinetic data structures. Maintain a set of certificates for each stable state: $$ c_1(t)>0, \ldots, c_m(t)>0. $$ When some: $$ c_j(t^\star)=0 $$ an event is triggered. Within an animation interval, there is therefore no need to reason through the entire topology again at every frame. It suffices to maintain: $$ \boxed{ \text{event certificates}. } $$ --- # 50. The Minimum Goal of Dynamic Projection Closure For a fixed path: $$ \Gamma, $$ if we can obtain: 1. finite stable intervals； 2. all critical event brackets； 3. each interval classification； 4. exact / certified transition types； 5. no unresolved time interval； it can be marked: $$ \boxed{ \texttt{PATH-CLOSED}. } $$ --- # 51. Family Closure If a family of paths: $$ \Gamma_\lambda, \qquad \lambda\in\Lambda $$ all have path certificates, this still does not mean the family is closed. We must additionally prove: $$ \boxed{ \Lambda = \bigcup_{i=1}^{N}\Lambda_i } $$ and control the event graph of every parameter cell: $$ \Lambda_i $$ . Only then do we have: $$ \boxed{ \texttt{FAMILY-CLOSED}. } $$ --- # 52. Why This Is Especially Important for Geometry Problems The gaps in many geometric conjectures lie not in ordinary states, but in: $$ \boxed{ \text{transition states}. } $$ For example: - exactly tangent； - exactly coplanar； - exactly collinear； - exactly support-tied； - exactly visibility-tied； - exactly at topology switch。 If only stable states are considered, these states are easily overlooked. --- # 53. Lebesgue Geometry as a Future Case Study Problems of Lebesgue universal covering involve: - rotation； - translation； - support function； - contact sequence； - convex hull； - witness placement； - topology wall； and are therefore naturally suited to Dynamic Projection Geometry. This paper does not, however, depend on Lebesgue. More generally, whenever a problem has: $$ \boxed{ \text{continuous geometric degrees of freedom} } $$ and: $$ \boxed{ \text{projection / reduction to lower-dimensional observable states}, } $$ the framework in this paper can be used. --- # 54. Possible Uses in Lebesgue Geometry In future, a three-dimensional source body: $$ M $$ can be built so that: - witness； - container； - support plane； - projection plane； - camera； can move relative to one another. At each instant, project: $$ G_t $$ back into two dimensions. The system computes simultaneously: - convexity； - width； - support signature； - contact graph； - area； - covering status； - branch label。 Any: $$ \texttt{UNKNOWN} $$ in the animation immediately becomes a counterexample candidate. --- # 55. Three-Dimensional Projection Cannot Replace a Planar Proof This must be stated explicitly. Even if: $$ 10^{12} $$ animation paths all fail to find a counterexample, this still does not directly imply: $$ \boxed{ \forall G\in\mathcal A, P(G). } $$ Animation stress testing supplies: $$ \boxed{ \text{adversarial evidence} } $$ not an unconditional global theorem. Actual global closure still requires: - exhaustive parameter cover； - analytic reduction； - interval certificate； - formal argument； - or another provable completeness mechanism. --- # 56. Which Layer Animation Can Actually Supplement Its greatest strengths are: $$ \boxed{ \text{discovering missing branches} } $$ and: $$ \boxed{ \text{exposing transition geometries}. } $$ It is therefore particularly suited to the: $$ \text{near-closure} $$ stage. When mathematics already regards: $$ \mathcal B = \{B_1,\ldots,B_n\} $$ as nearly complete, use Dynamic Projection Geometry deliberately to try to generate: $$ B_{n+1}. $$ --- # 57. Failure Is Success During the final stress test, if animation finds: $$ G^\star \notin \bigcup_iB_i, $$ this is not a system failure. Instead: $$ \boxed{ \text{the stress test succeeded}. } $$ because it prevents false closure. --- # 58. Replayability Every important animation should retain not just: ```text .mp4 ``` but: ```text source geometry initial state control path projection settings event ledger critical states certificate ``` That is: $$ \boxed{ \text{video is evidence rendering, not canonical experiment state}. } $$ --- # 59. Canonical Animation Source A formal research artifact can use: ```text scene.json motion.json projection.json events.jsonl certificate.json ``` together with: ```text README.md ``` Blender project： ```text .blend ``` can serve as a working carrier, but should not be the sole canonical mathematical source. --- # 60. Cross-Runtime Replay A mature system should permit: $$ \boxed{ \text{Blender} \rightarrow \text{independent geometry runtime} } $$ to replay the same: $$ \Gamma. $$ Alternatively: $$ \boxed{ \text{Runtime A} \leftrightarrow \text{Runtime B}. } $$ If two runtimes disagree on event ordering, mark: ```text CROSS-RUNTIME DISAGREEMENT ``` --- # 61. Animated Interaction and Certificate Obligations The certificate obligation for animated interaction should not be: > The image looks continuous. Instead, it should be: 1. the source state is replayable; 2. the transform path is explicit; 3. the projection operator is explicit; 4. the event functions are explicit; 5. the event intervals are verifiable; 6. unresolved intervals are explicit; 7. critical states can be reconstructed. --- # 62. Dynamic Projection Research Loop The paper recommends the complete loop: $$ \boxed{ \text{Build} \rightarrow \text{Move} \rightarrow \text{Project} \rightarrow \text{Observe} \rightarrow \text{Detect} \rightarrow \text{Freeze} \rightarrow \text{Verify} \rightarrow \text{Classify} \rightarrow \text{Continue}. } $$ --- # 63. Relation to Paper 01 Paper 01 establishes the core claim: $$ \boxed{ 3D \text{ can serve as an expansion space for 2D possibilities}. } $$ Paper 02 further asks: > How should this expansion space be swept continuously? The answer is: $$ \boxed{ \text{animation interaction}. } $$ --- # 64. Relation to the Next Paper This paper currently keeps: - who is observing; - what the object is; - which boundary / interface determines the projection; inside relatively abstract operators. The next paper will explicitly separate: $$ \boxed{ S = \text{Subject}, } $$ $$ \boxed{ O = \text{Object}, } $$ $$ \boxed{ \partial = \text{Boundary / Interface}. } $$ The next paper is therefore: $$ \boxed{ \textbf{Paper 03 — Subject–Object–Boundary Geometry}. } $$ --- # 65. Summary of the Paper's Main Propositions ## Proposition A $$ \boxed{ \text{Animation can be treated as a continuous path in geometric configuration space.} } $$ ## Proposition B $$ \boxed{ \text{3D motion can generate 2D configurations not naturally exposed by direct static 2D enumeration.} } $$ ## Proposition C $$ \boxed{ \text{2.5D is a provenance-preserving intermediate state, not merely a rendering style.} } $$ ## Proposition D $$ \boxed{ \text{critical projection events must be isolated independently of ordinary frame sampling.} } $$ ## Proposition E $$ \boxed{ \text{interactive animation can operate as a counterexample-search interface.} } $$ ## Proposition F $$ \boxed{ \text{visual replay is weaker than a Dynamic Projection Certificate.} } $$ --- # 66. Non-Claims This paper does not claim: 1. that all two-dimensional geometry problems should be lifted to three dimensions; 2. that every three-dimensional projection is more effective than direct two-dimensional analysis; 3. that one animation can prove closure of an infinite configuration space; 4. that Blender or a game engine can directly serve as a theorem prover; 5. that finite FPS guarantees that no critical events are missed; 6. that every event set is necessarily finite; 7. that 2.5D is the only intermediate representation; 8. that failure to find a counterexample in animation means a theorem holds; 9. that interaction can replace interval arithmetic or formal proof; 10. that the Lebesgue problem has already been closed by this framework. --- # 67. Conclusion The paper redefines animation, from a visual presentation, as: $$ \boxed{ \textbf{continuous geometric experimentation}. } $$ The three-dimensional source body: $$ M $$ is not the final object of study. It is a configuration generator with more degrees of freedom. Through: $$ T_t $$ and: $$ \Pi_t, $$ we obtain: $$ \boxed{ G_t = \Pi_t(T_tM). } $$ What genuinely matters is: $$ \boxed{ \text{which geometric types and critical events the entire }G_t\text{ path traverses, and whether it generates unknown classes.} } $$ Therefore: $$ 3D \rightarrow 2.5D \rightarrow 2D $$ is not just a one-way dimensional reduction. It is an architecture of: $$ \boxed{ \text{possibility generation} + \text{provenance retention} + \text{event detection} + \text{counterexample search} } $$ . Ultimately, the research value of animated interaction does not lie in: > Can we animate the geometry problem? but in: $$ \boxed{ \text{Can continuous animation actually reveal possibilities invisible to static mathematical enumeration?} } $$ If the answer is yes, animation is no longer merely the final presentation of mathematical results. It can become: $$ \boxed{ \text{a geometric attack layer before final closure.} } $$ --- # Canonical Source Declaration This file is the UTF-8 Markdown canonical source for `PPS-DPG-02-v0.1`. The only mathematical delimiters used are `$...$` and `$$...$$`. No other LaTeX display / inline delimiter is used as a canonical source delimiter. The paper defines animated interaction as continuous geometric experimentation and strictly distinguishes: - 3D source state； - 2.5D provenance state； - 2D projection state； - render output； - event certificate； - global mathematical closure。 ===END===