Coordinate Conversion Tool
Enter a Cartesian point and define the lattice. The tool solves the inverse lattice matrix.
Formula Used
Let the Cartesian point be r = [x, y, z]. Let the origin offset be o. Let the lattice matrix be A = [a b c], where each column is a lattice vector. The fractional coordinate vector is f = A⁻¹(r - o).
For cell parameters, the tool first builds the lattice matrix. It uses lengths a, b, c and angles alpha, beta, gamma. Then it applies the same inverse matrix formula.
How to Use This Calculator
- Enter the Cartesian x, y, and z coordinates.
- Choose the unit used for the Cartesian point.
- Enter an origin offset when your model has one.
- Select lattice vectors or cell parameters.
- Enter vector components or cell lengths and angles.
- Choose output precision and fraction denominator limits.
- Enable wrapping when you need values inside one unit cell.
- Press the calculate button to view the result.
Example Data Table
| Cartesian Point | Lattice | Expected Fractional Result |
|---|---|---|
| (0.25, 0.50, 0.75) | a=(1,0,0), b=(0,1,0), c=(0,0,1) | (0.25, 0.50, 0.75) |
| (2, 3, 4) | a=(4,0,0), b=(0,6,0), c=(0,0,8) | (0.50, 0.50, 0.50) |
| (1, 1, 0) | a=(2,0,0), b=(1,2,0), c=(0,0,3) | (0.25, 0.50, 0) |
Understanding Cartesian and Fractional Coordinates
Cartesian coordinates describe a point with fixed axis distances. They use x, y, and z values. Fractional coordinates describe the same point differently. They measure position along lattice vectors. This makes them useful in crystal structures. A value of 0.5 means halfway along that vector.
The conversion needs a lattice basis. The basis may come from three vectors. It may also come from cell lengths and angles. The calculator accepts both methods. That helps users work with different data sources. Many files list cell parameters. Other modeling tools export full vectors. Both formats can define the same cell.
Why Fractional Coordinates Matter
Fractional values are compact and cell aware. They stay meaningful when a cell changes size. They also make symmetry operations easier. Many crystal formats store atoms this way. Researchers can compare structures with fewer unit problems. The method also helps when atoms cross cell boundaries.
Cartesian coordinates are still important. They show real distances in a chosen unit. They are helpful for plotting and measuring geometry. Yet they do not directly show the unit cell position. Fractional coordinates solve that issue. They connect the point to the repeating lattice.
Handling Non Orthogonal Cells
Many cells are not cubic. Some have tilted axes. Angles may differ from ninety degrees. That makes manual conversion harder. The inverse matrix method handles these cells. It uses the full geometry of the basis. This is why vector input is powerful. It avoids assumptions about square axes.
When using cell parameters, angle quality matters. Invalid angles can create impossible cells. Very small volume can also cause unstable results. The calculator checks the determinant. A near zero determinant means the vectors are not independent. In that case, no unique fractional result exists.
Accuracy Tips
Good input data improves every result. Use the same coordinate reference for all points. Check exported units from every source before conversion. Some programs store nanometers. Others store angstrom values. Mixed units create large errors. Also check whether coordinates include an origin shift. A shifted origin changes all fractional values. For batches, keep one lattice definition. Then compare points under consistent geometry. Use the residual check as a warning. Precision should match source quality. Extra decimals cannot fix uncertain measurements.
Origins, Units, and Wrapping
Some coordinate sets use a shifted origin. The tool subtracts that offset first. Units are also converted before solving. This prevents mixed scale errors. Wrapping is optional. It moves values into one unit cell. For example, 1.25 becomes 0.25. That point is equivalent in a repeated lattice.
Fraction outputs are useful for ideal positions. Many atoms appear near simple fractions. A decimal result near one third may show as 1/3. The denominator limit controls this estimate. Use higher limits for detailed cases. Use lower limits for cleaner crystallographic patterns. Clear fractional values make crystal comparisons faster and safer.
FAQs
1. What are fractional coordinates?
Fractional coordinates describe a point relative to lattice vectors. The values show how far the point lies along a, b, and c directions.
2. What is the main conversion formula?
The calculator uses f = A⁻¹(r - o). Here A is the lattice matrix, r is the Cartesian point, and o is the origin offset.
3. Can I use cell lengths and angles?
Yes. Choose the cell parameter mode. Enter a, b, c, alpha, beta, and gamma. The calculator builds the lattice matrix first.
4. What does wrapping do?
Wrapping moves each fractional value into the 0 to 1 range. It is useful for points outside the displayed unit cell.
5. Why do I need lattice vectors?
Lattice vectors define the cell shape and scale. Without them, the same Cartesian point could mean many different fractional positions.
6. What if the determinant is zero?
A zero determinant means the vectors are dependent. The cell has no usable volume, so a unique fractional coordinate cannot be found.
7. Which units can I use?
You can use angstrom, nanometer, picometer, or meter. The calculator converts compatible inputs before applying the matrix operation.
8. Why is there a fraction result?
The fraction result gives a clean rational estimate. It helps identify common positions, such as 1/2, 1/3, or 1/4.
9. Can this handle tilted cells?
Yes. The inverse matrix method works for non orthogonal cells. Enter accurate vectors or valid cell angles for best results.
10. What is the origin offset?
The origin offset is a shift applied before conversion. Use it when your Cartesian point is measured from a different origin.
11. Are wrapped coordinates different points?
They keep the same point within a repeated lattice.