📐 Corner Point Calculator
Find corner points of a feasible region in linear programming
How to Use This Calculator
Enter Constraints
Input your linear constraints in the form ax + by ≤ c (or ≥). Include x ≥ 0, y ≥ 0 if needed.
Calculate
Click to find all corner points (intersections of constraint lines).
Use for Optimization
In linear programming, optimal solutions occur at corner points. Evaluate your objective function at each corner point.
Formula
Corner Points = Intersections of Constraint Lines
For constraints: a₁x + b₁y ≤ c₁, a₂x + b₂y ≤ c₂
Finding Intersections:
Solve system: a₁x + b₁y = c₁, a₂x + b₂y = c₂
x = (c₁b₂ - c₂b₁) / (a₁b₂ - a₂b₁)
y = (a₁c₂ - a₂c₁) / (a₁b₂ - a₂b₁)
Feasible Region:
Set of all points (x, y) satisfying all constraints simultaneously
Corner Point Theorem:
In linear programming, if an optimal solution exists, it occurs at a corner point of the feasible region.
About Corner Point Calculator
The Corner Point Calculator finds the corner points (vertices) of a feasible region defined by linear constraints. These points are intersections of constraint lines and are crucial in linear programming optimization problems.
When to Use This Calculator
- Linear Programming: Find corner points for simplex method
- Optimization: Locate candidate optimal solutions
- Feasible Region: Understand geometry of constraints
- Operations Research: Resource allocation problems
Why Use Our Calculator?
- ✅ All Corner Points: Finds all intersections
- ✅ Feasibility Check: Validates points against constraints
- ✅ Flexible: Supports ≤ and ≥ constraints
- ✅ Educational: Helps understand LP geometry
- ✅ Free: No registration required
Key Concepts
- Corner Point: Vertex of the feasible region polygon
- Feasible Region: Set of all points satisfying constraints
- Fundamental Theorem: Optimal LP solution at corner point (if it exists)
- Bounded vs Unbounded: Feasible region can be bounded or extend infinitely
Frequently Asked Questions
What are corner points?
Corner points are vertices of the feasible region polygon - intersections where constraint lines meet. In linear programming, optimal solutions occur at corner points.
Why are corner points important?
The fundamental theorem of linear programming states that if an optimal solution exists, it occurs at a corner point. This allows checking only a finite number of points instead of infinitely many.
What if no corner points are found?
This could mean: (1) Constraints are inconsistent (no feasible region), (2) Region is unbounded, or (3) All constraints are redundant. Check your constraint setup.
Do I need to include x ≥ 0, y ≥ 0?
Only if your problem requires non-negativity constraints. Many LP problems assume x ≥ 0, y ≥ 0 implicitly. Add them as separate constraints if needed.