Calculator Inputs
Example Data Table
| Example | Set | Test point | Boundary result | Reason |
|---|---|---|---|---|
| 1 | Interval [2, 9] | x = 2 | Yes | The point matches the left endpoint. |
| 2 | Rectangle [0, 6] x [0, 4] | (6, 1.5) | Yes | The point lies on the right edge. |
| 3 | Circle center (0, 0), radius 5 | (3, 4) | Yes | The distance from the center equals the radius. |
| 4 | Interval [2, 9] | x = 5 | No | The point is interior, not an endpoint. |
Formula Used
A boundary point of a set sits where every small neighborhood intersects both the set and its complement. This calculator uses direct formulas for common sets.
- Interval [a, b]: the boundary is
{a, b}. A point is on the boundary whenx = aorx = b, within tolerance. - Rectangle [xmin, xmax] x [ymin, ymax]: a point is on the boundary when it lies on at least one edge and still stays within the rectangle limits.
- Circle: compute
d = sqrt((x - h)^2 + (y - k)^2). The point is on the boundary when|d - r| <= tolerance. - Nearest boundary distance: the calculator also reports the smallest distance from the test point to an endpoint, edge, or circular boundary condition.
How to Use This Calculator
- Select the set or shape you want to analyze.
- Enter the interval limits, rectangle bounds, or circle center and radius.
- Type the coordinates of the test point.
- Set a tolerance if you want approximate comparison.
- Press Calculate Boundary Point to show the result above the form.
- Review the classification, membership checks, and distance outputs.
- Use the CSV or PDF buttons to save the current result.
Frequently Asked Questions
1. What is a boundary point?
A boundary point is a location where every tiny neighborhood touches the set and also touches points outside the set. It marks the edge between membership and non-membership.
2. Why does the calculator use tolerance?
Many coordinates come from rounded values or measurements. Tolerance lets the calculator treat very small numerical differences as equal, which is useful for practical computations.
3. Are interval endpoints always boundary points?
For a finite interval in the real line, the endpoints form the boundary. Interior points are not boundary points because nearby values can remain fully inside the interval.
4. How is the rectangle boundary checked?
The calculator checks whether the point lies on the left, right, bottom, or top edge while staying within the rectangle's overall coordinate range.
5. How is the circle boundary checked?
It calculates the distance from the point to the center. If that distance matches the radius within tolerance, the point is classified as a boundary point.
6. What does closure membership mean here?
Closure membership tells you whether the point belongs to the set or sits on its limit under the chosen tolerance. It helps separate near-edge cases from clear exterior points.
7. Can this help with topology homework?
Yes. It gives concrete examples for standard sets, shows formulas, and helps you compare interior, boundary, and exterior classifications quickly.
8. What does nearest boundary distance show?
It measures how far the test point is from the closest edge or endpoint. Small values indicate the point is close to the boundary, even if not exactly on it.