Calculator Form
Formula Used
The calculator starts with two points, A(x₁, y₁) and B(x₂, y₂). It measures their coordinate differences.
dx = x₂ - x₁
dy = y₂ - y₁
d = √(dx² + dy²)
E = w(dx² + dy²)
Here, w is the optional E weight coefficient. The transformed coordinates use rotation, scale, and translation.
x′ = s(x cosθ - y sinθ) + tx
y′ = s(x sinθ + y cosθ) + ty
The raw transformed E changes when scale is not one. Therefore, the normalized transformed value is:
Normalized E′ = Raw E′ ÷ s²
The invariant error is the absolute difference between original E and normalized transformed E.
How to Use This Calculator
- Enter the first point coordinates in x₁ and y₁.
- Enter the second point coordinates in x₂ and y₂.
- Add translation values for the transformed plane.
- Enter a rotation angle and choose degrees or radians.
- Enter a uniform scale value greater than zero.
- Set the E weight coefficient and tolerance.
- Press the calculate button to view the result above the form.
- Use the CSV or PDF button to export the result.
Example Data Table
| x₁ | y₁ | x₂ | y₂ | Rotation | Scale | Weight | Expected E |
|---|---|---|---|---|---|---|---|
| 2 | 3 | 8 | 11 | 35° | 1.5 | 1 | 100 |
| -4 | 6 | 5 | -2 | 90° | 2 | 0.5 | 72.5 |
| 1.25 | -3.5 | 9.75 | 4 | 0.785 rad | 0.8 | 2 | 257 |
Article: Understanding Invariant Two0-Dimensional E
Purpose of the Measure
Invariant two0-dimensional E is a practical way to test whether a paired point measurement stays stable after a geometric change. The calculator treats two points as a segment in a flat plane. It then measures the squared length of that segment. A weight can be added when the segment represents intensity, cost, energy, or another scaled quantity.
Why Invariance Matters
Many models move shapes without changing their true internal size. A map may shift. A drawing may rotate. A sensor frame may use another origin. In these cases, the relation between two points should remain the same. The invariant E value helps confirm that relation. It ignores translation. It also handles rotation. With normalization, it can compare scaled versions too.
How the Calculator Works
The tool first calculates dx and dy. These values describe the horizontal and vertical difference between the two points. It squares both values and adds them. This gives the squared distance. The selected weight multiplies that amount. The result is the original E value.
Transformation Process
Next, the calculator transforms both points. It applies rotation around the origin. It multiplies coordinates by a uniform scale. It then adds translation values. The transformed points create a second segment. Its raw E may increase or decrease when scale is not one. For that reason, the tool divides raw transformed E by scale squared.
Error and Tolerance
The final comparison uses absolute error. A small error usually comes from rounding. The tolerance field lets you define the acceptable difference. If the error is within that limit, the calculator reports that the invariant condition passes. This is useful for checking geometry, graphics, coordinate conversions, layout testing, and simple modeling tasks.
Good Input Practices
Use consistent units for all coordinates. Avoid a zero scale. Pick a realistic tolerance for your data. More decimal places can help when checking small differences. A larger weight makes E larger, but it does not change the invariant principle.
FAQs
1. What does invariant E mean here?
It means a weighted squared distance between two points. The value should stay unchanged after rotation and translation. It can also stay comparable after scaling when normalized by scale squared.
2. Why does the calculator use squared distance?
Squared distance avoids the square root step and gives a stable E value. It is useful for geometry checks, model comparisons, and coordinate transformation testing.
3. Can I use negative coordinates?
Yes. Negative coordinates work normally. The calculator squares coordinate differences, so direction does not make the final E negative.
4. What does the weight coefficient do?
The weight multiplies the squared distance. Use it when the distance represents a stronger or weaker effect in your model.
5. Why is scale not allowed to be zero?
A zero scale collapses all points into one location. Normalized E also divides by scale squared, so zero would make the formula invalid.
6. What tolerance should I choose?
Use a small tolerance for precise data. Use a larger tolerance when inputs are rounded or measured with lower accuracy.
7. Does translation change invariant E?
No. Translation moves both points by the same amount. Their difference remains unchanged, so original E stays the same.
8. What export options are included?
The calculator includes CSV and PDF exports. They save input values, transformed values, E results, errors, and status.