Scope: This tool actively moves Cartesian points and shapes in a fixed 2D plane. It is not a latitude–longitude or geographic datum converter, a passive change of coordinate basis, or a 3D transformation tool.
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Accepts 1,2, (1, 2), 1 2, tab-separated cells, several parenthesized pairs on one line, or CSV with an optional header.
Operations run from top to bottom. Drag a step or use the arrow buttons to reorder it.
Commands: translate dx dy; rotate angle cx cy; scale sx sy cx cy; shear horizontal|vertical k; reflect x|y|origin|y=x|y=-x.
Browser calculations use floating-point arithmetic. Exact quarter-turn rules prevent tiny artifacts around 90°, 180° and 270°.
Expected: P′=(-1,3).
Expected: A′=(0,0), B′=(1,2), C′=(0,4).
Scale 2: P=(3,2) becomes P′=(5,3).
For P=(1,0), translate then rotate gives (0,3); reverse the steps to compare.
Enter x and y, choose the operation, and substitute them into its coordinate rule. The answer is named P′, the image of P.
Enter every vertex in boundary order. Apply the same rule independently to A, B, C and the other vertices; close the graph for a polygon.
Translations and rotations preserve distance, angle and orientation. Reflections preserve distance and angle but reverse orientation. Uniform dilations preserve angle and scale lengths by |k|. Shears preserve parallel lines.
A constant coordinate difference suggests translation; equal radii suggest rotation; equal perpendicular distances from a line suggest reflection; proportional center distances suggest dilation.
Translate by the opposite vector, rotate by the opposite angle, reflect over the same line, dilate by reciprocal nonzero factors, or apply sequence steps in reverse with each inverse.
(x,y)→(−y,x)(x,y)→(y,−x)(x,y)→(−x,−y)(x,y)→(x,−y)(x,y)→(−x,y)(x,y)→(−x,−y)(x,y)→(y,x)(x,y)→(−y,−x)(2k−x,y) / (x,2k−y)Start with P=(1,0). Translate by (2,0) and then rotate 90° counterclockwise: (1,0)→(3,0)→(0,3). Reverse the order: (1,0)→(0,1)→(2,1). The image changes because transformation composition is generally not commutative.
Using column vectors, [x′,y′,1]ᵀ=M[x,y,1]ᵀ. Translation uses [[1,0,tₓ],[0,1,tᵧ],[0,0,1]]; rotation uses [[cosθ,−sinθ,0],[sinθ,cosθ,0],[0,0,1]]; dilation uses [[sₓ,0,0],[0,sᵧ,0],[0,0,1]]; horizontal shear uses [[1,k,0],[0,1,0],[0,0,1]]. For operations in listed order, M=Mₙ…M₂M₁.
Authorship and verification owner: Starlight Robotics, the calculator publisher and maintainer. Formula and interaction review completed August 31, 2026.
Method: Every operation is converted to a homogeneous 3×3 matrix, applied to the coordinate vector, and independently checked against known classroom rules. Automated self-tests cover identity transforms, a translation and its inverse, four quarter-turns, reflection twice, preserved rotation distance, dilation area factor logic, and horizontal/vertical shear rules. A failure is reported in the browser console without altering user results.
Numerics: JavaScript uses IEEE 754 floating-point arithmetic. Display rounding follows the selected precision; exact quarter-turn handling replaces near-zero trigonometric residues for standard 90° steps.
References: Gilbert Strang, Introduction to Linear Algebra; OpenStax, Precalculus 2e; David C. Lay, Steven R. Lay and Judi J. McDonald, Linear Algebra and Its Applications, 6th ed. No rating or testimonial is claimed.
Choose Rotate, select clockwise, and enter a positive angle. The calculator applies the equivalent negative counterclockwise angle; for example, 90° clockwise maps (x,y) to (y,−x) about the origin.
For the vertical line x=k, use (x′,y′)=(2k−x,y). For the horizontal line y=k, use (x′,y′)=(x,2k−y). Choose either mirror line directly and enter k.
A negative scale factor places the image on the opposite side of the dilation center. Its absolute value controls size; −1 produces a half-turn about the center.
A uniform dilation multiplies every length by |k|. Area has two dimensions, so it is multiplied by |k|×|k|=k².
Choose Multiple points or shape, enter every vertex in boundary order, and enable Close shape. The same rule is applied to A, B, C and every remaining vertex.
Not usually. A nonzero translation does not keep the origin fixed, so it is affine rather than linear. Homogeneous 3×3 matrices represent it alongside linear transformations.