Hyperplane Equation Calculator

Calculator Input

Dimension

Choose the number of variables in the hyperplane.

Decimal Precision

This controls rounded output and export values.

Graph Span

Wider spans show more of the plotted region.

Normal Vector n

The normal vector defines orientation and coefficients.

Point on Hyperplane p

This point must lie on the desired hyperplane.

Test Point t

The calculator measures distance and side for this point.

Formula Used

For a normal vector n = (a₁, a₂, …, a_n) and a point p = (p₁, p₂, …, p_n), the hyperplane is:

n · (x - p) = 0

The expanded equation becomes:

a1x1 + a2x2 + ... + anxn + d = 0, where d = -(n · p)

The signed distance from a test point t to the hyperplane is:

(n · t + d) / ||n||

The orthogonal projection of the test point onto the hyperplane is:

proj(t) = t - ((n · t + d) / ||n||²)n

How to Use This Calculator

Select the dimension of the space, from 2 through 10.
Enter the normal vector components. At least one value must be nonzero.
Enter a point that lies on the hyperplane.
Provide any test point to measure distance and determine its side.
Choose decimal precision and graph span if you want customized output.
Press Calculate Hyperplane to show the equation above the form.
Review the equation, unit normal, projection point, intercepts, and graph.
Use the CSV or PDF buttons to save the results.

Example Data Table

Dimension	Normal Vector	Point on Hyperplane	Test Point	Expanded Equation	Signed Distance
3	(2, -3, 4)	(1, 2, -1)	(3, 0, 2)	2x1 - 3x2 + 4x3 + 8 = 0	4.0853

FAQs

1) What is a hyperplane?

A hyperplane is a flat geometric set in higher dimensions. In two dimensions it is a line, and in three dimensions it is a plane.

2) Why do I need a normal vector?

The normal vector gives the direction perpendicular to the hyperplane. Its components also become the main coefficients in the expanded equation.

3) What does the point on the hyperplane do?

The point anchors the hyperplane in space. Without a point, the normal vector only tells orientation, not location.

4) What is the meaning of the constant term d?

The constant term shifts the hyperplane so it passes through the chosen point. It is computed as the negative dot product of the normal vector and point.

5) What does signed distance tell me?

Signed distance measures how far the test point is from the hyperplane. Its sign also tells which side of the hyperplane the point lies on.

6) Can this calculator work beyond three dimensions?

Yes. The equation, distance, projection, and intercept calculations work in higher dimensions. The graph becomes a summary chart because direct plotting is not possible above three dimensions.

7) Why are some intercepts missing?

An intercept for a coordinate is unavailable when its coefficient is zero. In that case, the hyperplane does not cross that axis in the simple intercept form.

8) What happens if the normal vector is zero?

A zero normal vector cannot define a hyperplane. The calculator blocks this case and asks for at least one nonzero coefficient.

Advanced Hyperplane Equation Calculator