Calculator
Formula used
The calculator transforms both input points first. The applied order is scale, rotate, and translate.
x_s = x × sx and y_s = y × sy
x_r = x_s cos θ - y_s sin θ
y_r = x_s sin θ + y_s cos θ
x' = x_r + tx and y' = y_r + ty
For transformed points P1(x1', y1') and P2(x2', y2'), the general line form is:
A = y1' - y2'
B = x2' - x1'
C = x1'y2' - x2'y1'
Ax + By + C = 0
How to use this calculator
- Enter the first point as X and Y values.
- Enter the second point as X and Y values.
- Add translation values if the line must be shifted.
- Add scale values if coordinates must be stretched.
- Add the rotation angle in degrees.
- Choose the decimal precision for the final result.
- Press calculate to see the transformed points and line equation.
- Use CSV or PDF buttons to save the result.
Example data table
| Point 1 | Point 2 | Translation | Scale | Rotation | General form result |
|---|---|---|---|---|---|
| (0, 0) | (4, 2) | (0, 0) | (1, 1) | 0° | -x + 2y = 0 |
| (0, 0) | (4, 2) | (1, 3) | (1, 1) | 0° | -x + 2y - 5 = 0 |
| (1, 0) | (1, 3) | (0, 0) | (1, 1) | 90° | y - 1 = 0 |
Understanding the transformed line
A line can be created from any two distinct points. This calculator first transforms the points. Then it builds the line through the new coordinates. The result is written in general form. General form is Ax + By + C = 0. It is useful because it works for vertical lines, horizontal lines, and slanted lines.
Why transformations matter
Point transformations change the position of a shape before the final equation is created. Translation moves every point by the same amount. Scaling stretches the coordinates along each axis. Rotation turns the scaled points around the origin. These steps help model graphics, mapping, analytic geometry, and classroom coordinate problems. The calculator applies the steps in a clear order. It scales first, rotates second, and translates last.
What the result shows
The output includes the transformed coordinates. It also shows A, B, and C. These values define the final line. The tool also reports slope when the line is not vertical. It gives x intercept and y intercept when they exist. A normalized version can be produced for cleaner comparison. The distance between transformed points is included as a quality check. The line angle helps connect the equation with direction.
Using the tool well
Enter two different starting points. Add translation, scale, and rotation values as needed. Use one for scale when no stretching is needed. Use zero for rotation when no turn is needed. Select the precision level that matches your assignment. Then press calculate. Review the transformed points first. If they are identical, no unique line exists.
Practical uses
This calculator is helpful for geometry lessons and coordinate transformation work. It can support CAD checks, graphing tasks, game math, and tutoring examples. The CSV file is useful for spreadsheets. The PDF file is useful for reports. The example table shows common input patterns. You can compare those cases with your own values. This makes the method easier to verify and reuse.
Accuracy notes
Floating results may look slightly long after rotation. That is normal for decimal trigonometry. Choose fewer decimals for display. Choose more decimals when checking exact input work or comparing separate transformation steps in careful geometry reviews.
FAQs
What is line general form?
Line general form is Ax + By + C = 0. It stores a line using three coefficients and works for vertical, horizontal, and slanted lines.
Can this calculator handle vertical lines?
Yes. General form handles vertical lines safely. The slope will show as undefined, but the equation is still valid.
What transformation order is used?
The calculator scales the points first. It then rotates them around the origin. Finally, it translates the rotated points.
What happens when transformed points match?
No unique line exists when both transformed points become identical. The calculator shows an error so you can change the inputs.
Why should I normalize coefficients?
Normalization divides A, B, and C by the line vector length. It makes coefficient comparisons cleaner and easier to read.
Can I use negative scale values?
Yes. Negative scale values can reflect coordinates across an axis before rotation and translation are applied.
What does the CSV download include?
The CSV file includes inputs, transformed points, coefficients, equation, slope, intercepts, and distance. It is ready for spreadsheets.
What does the PDF download include?
The PDF file gives a clean report of the same solved values. It is useful for homework, records, and geometry notes.