Rotation Around Y Axis Calculator

Calculator Input

X Coordinate

Y Coordinate

Z Coordinate

Rotation Angle

Angle Unit

Direction

Pivot X

Pivot Y

Pivot Z

Decimal Places

Formula Used

For a point P(x, y, z), subtract the pivot C(a, b, c). This gives a shifted point.

X = x - a, Y = y - b, Z = z - c

The y axis rotation matrix is:

x′ = X cosθ + Z sinθ
y′ = Y
z′ = -X sinθ + Z cosθ

Then add the pivot back:

Final point = (x′ + a, y′ + b, z′ + c)

How to Use This Calculator

Enter the original x, y, and z coordinates.
Enter the rotation angle.
Choose degrees or radians.
Select clockwise or counterclockwise direction.
Enter pivot values if rotation is not around the origin.
Choose decimal places for the final answer.
Click the calculate button.
Download CSV or PDF after the result appears.

Example Data Table

X	Y	Z	Angle	Unit	Direction	Pivot	Expected Idea
3	2	5	90	Degrees	Counterclockwise	(0, 0, 0)	x and z rotate in the xz plane.
4	1	0	180	Degrees	Counterclockwise	(0, 0, 0)	x changes sign around the origin.
6	3	2	45	Degrees	Clockwise	(1, 0, 1)	The point rotates around a shifted vertical line.

What Is Rotation Around the Y Axis?

A rotation around the y axis turns a 3D point through a chosen angle while keeping its vertical coordinate unchanged. It is common in analytic geometry, graphics, robotics, and game design. The calculator applies the standard right handed matrix, so positive angles follow the usual mathematical convention.

Why This Calculator Helps

Manual rotation can become confusing when signs, radians, degrees, and pivots are involved. This tool accepts coordinates, angle units, direction, and a custom pivot. It then shows the shifted point, matrix values, rotated coordinates, radius from the y axis, and projected angle in the xz plane. These details make each step easier to check.

Formula Logic

The point is first shifted by subtracting the pivot coordinates. The rotated x value uses cosine with the old x value and sine with the old z value. The rotated z value uses negative sine with x and cosine with z. The y value stays the same during the rotation. Finally, the pivot is added back to the result. This gives coordinates in the original space.

Practical Uses

Students can use the calculator to test homework answers. Designers can rotate model points around a vertical axis. Developers can confirm transformation code before adding it to larger systems. Teachers can build examples that compare clockwise and counterclockwise rotation.

Accuracy Tips

Use enough decimal places for small angles or large coordinates. Choose radians only when your angle is already in radians. Check the direction field before exporting your result. For rotations around the origin, leave the pivot values as zero. For rotations around another vertical line, enter the pivot x and z values. The pivot y value is included for completeness, although y remains unchanged relative to that pivot.

Reading Results

The final coordinates show the new point after rotation. The matrix section shows the exact trigonometric coefficients used. The distance from the y axis should stay constant when the pivot is fixed. The inverse check rotates the answer back by the opposite angle. If it matches the original point, the calculation is consistent.

Export Options

Use the CSV button for spreadsheets. Use the PDF button for reports. Both downloads keep the entered values and computed coordinates together.

FAQs

What does rotation around the y axis mean?

It means turning a 3D point around a vertical y axis. The y coordinate stays unchanged relative to the pivot. The x and z coordinates move in the horizontal plane.

Does the calculator support radians?

Yes. You can choose degrees or radians. Select radians only when your angle value is already measured in radians.

What is the default pivot?

The default pivot is the origin, written as (0, 0, 0). You can enter another pivot to rotate around a shifted vertical line.

Why does y stay the same?

A y axis rotation moves the point around the vertical axis. That motion changes x and z values, but not the vertical distance from the pivot.

What direction is positive rotation?

The calculator uses the standard right handed convention. Counterclockwise uses the entered angle as positive. Clockwise uses the negative angle.

Can I download the result?

Yes. After calculating, you can download a CSV file or a PDF file. Both include entered values and computed coordinates.

What is the inverse check?

The inverse check rotates the final point back by the opposite angle. It should return values very close to the original coordinates.

Why do tiny decimal differences appear?

Trigonometric calculations use floating point numbers. Very small differences can appear after rounding. Increase decimal places when you need more detail.