Solve nearest-point problems fast for structured geometric constraints. Choose vector inputs and inspect projection behavior. Clean layouts, exports, formulas, and graphs support deeper learning.
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| Set Type | Input x | Parameters | Projected Point | Notes |
|---|---|---|---|---|
| Axis-Aligned Box | (5, -1, 8) | l = (0, 0, 0), u = (4, 3, 6) | (4, 0, 6) | Each coordinate is clipped independently. |
| Euclidean Ball | (6, 8) | Center = (0, 0), radius = 5 | (3, 4) | Outside points shrink radially to the boundary. |
| Probability Simplex | (0.2, 0.3, 0.7) | Sum = 1 | (0.1333, 0.2333, 0.6333) | Coordinates stay nonnegative and sum to one. |
| Half-Space | (5, 4) | a = (1, 1), b = 6 | (3.5, 2.5) | Projection moves only when the inequality is violated. |
| Affine Hyperplane | (3, 1) | a = (1, -1), b = 0 | (2, 2) | The result lands exactly on the equality surface. |
For a convex set C, the projection of x is the nearest feasible point:
PC(x) = argminy ∈ C ||y - x||₂.
If C = {y : lᵢ ≤ yᵢ ≤ uᵢ}, then each coordinate is projected by clipping:
pᵢ = min(max(xᵢ, lᵢ), uᵢ).
If C = {y : ||y - c||₂ ≤ r}, keep x when it is inside. Otherwise use
p = c + r (x - c) / ||x - c||₂.
For C = {y : yᵢ ≥ 0, Σyᵢ = s}, the projection uses a threshold:
pᵢ = max(xᵢ - θ, 0), where θ is chosen so Σpᵢ = s.
For C = {y : aᵀy ≤ b}, keep x if feasible. Otherwise use
p = x - ((aᵀx - b) / ||a||₂²) a.
For C = {y : aᵀy = b}, project directly by
p = x - ((aᵀx - b) / ||a||₂²) a.
It means finding the closest point in a convex set to your input vector. The result minimizes Euclidean distance while satisfying the set’s geometric constraints.
Projection appears in constrained optimization, signal processing, machine learning, feasibility methods, and numerical algorithms. It converts an infeasible point into the nearest feasible one.
This version supports axis-aligned boxes, Euclidean balls, probability simplices, half-spaces, and affine hyperplanes. These cover many standard projection problems in applied mathematics.
The projection equals the original vector, so the projection distance becomes zero. The feasibility check shows that no corrective movement was needed.
It measures how far the original point was from the feasible set. Larger values mean stronger constraint violation or greater correction to reach feasibility.
Box projection clips each coordinate independently. Simplex projection must also keep all coordinates nonnegative while forcing their total sum to a chosen target.
Yes. Enter as many coordinates as needed, provided every related parameter vector has the same dimension. The graph compares coordinates rather than drawing full geometry.
Many projection problems are higher-dimensional, so a coordinate comparison stays informative for any dimension. It highlights which coordinates changed and by how much.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.