Overview
Constraint projection adjusts positions, velocities, or generalized coordinates after an unconstrained update so constraints such as distances, contacts, or joints are satisfied more closely.
CoreSource checked
Constraint Projection
Enforces simulation constraints by projecting invalid state back onto the allowed manifold.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Simulation
- Difficulty
- Intermediate
- Family
- Constraint Solver
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 5
- Code example
- Ready
- Visualization
- Ready
Core Concepts
constraint errorprojection stepposition correctioniterative solver
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(constraints)
Average
O(iterations * constraints)
Worst
O(iterations * constraints)
Space
O(state)
When to Use
- Use when a simulation step can drift away from constraints and needs correction.
- Use when approximate constraint satisfaction is acceptable over iterative projection.
When Not to Use
- Avoid when exact constraint forces or energy behavior are required.
- Avoid when projection creates visible popping or violates important conservation laws.
What It Does
- Corrects simulated positions by projecting them back onto constraint-satisfying states
- Uses constraint error and projection step to guide the work.
- Produces updated state.
Use Cases
- Physics simulation
- Animation
- Interactive systems
Inputs
- Current state
- Forces or constraints
- Time step
Outputs
- Updated state
- Constraint corrections
- Collision results
Key Formulas (1)
Constraint correction
δx = −C(x) / ||gradient C(x)||2 · gradient C(x)
delta_x = -C(x) / ||gradient C(x)||^2 * gradient C(x)Moves positions to reduce constraint error.
Pseudocode
- 1.Advance unconstrained state
- 2.For each constraint, compute violation
- 3.Compute a correction direction and magnitude
- 4.Apply correction to affected state variables
- 5.Repeat until residuals are within tolerance or iteration limit
Canonical Example
- Input
- two particles connected by a distance constraint of length 1 but currently 1.2 apart
- Output
- particles moved toward each other to restore distance 1
- Explanation
- Projection reduces the constraint violation after the unconstrained update.
Implementation Notes
- Compute constraint violation explicitly.
- Apply limited corrections and iterate when needed.
- Track residual error to decide whether more iterations are required.
Common Pitfalls
- Large corrections can introduce jitter or energy artifacts.
- Constraint ordering can change the final projected state.
- Projection tolerances must balance stability and visual accuracy.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow state updates, constraints, and error correction over simulation steps.
Step 1 of 5: Advance unconstrained state
simulation-mechanismNotes
Projection is a stabilization tool; it should expose tolerance and iteration limits.
Sources (1)
What each reference supports for this entry.
Real-Time Collision Detection
BookCollision detection reference for the geometric or broad-phase/narrow-phase behavior of Constraint Projection.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Used bySequential Impulse Solver
Simulation · Core
Both correct constraint errors iteratively, though sequential impulses work primarily through velocity-level impulses.
Compare with thisRelationships
View full relationship pageUsed byPosition Based Dynamics
PBD uses projection to satisfy constraints on positions.
Often paired withVerlet Constraint Solver
Constraint projection is commonly iterated in Verlet-style solvers.
Related Algorithms
Tags
SimulationConstraint ProjectionPhysicsConstraint Solving