Coordinate systems: 2D, 3D, right-handed, Z-up

To describe where something is, you need three things: an origin (the zero point), a set of axes (directions to measure along) and a unit (how long “1” is). A coordinate is just the distance from the origin along each axis.

In plan: X and Y

On a floor plan, the usual convention is X to the right (east on a north-up drawing) and Y up the page (north). A column at (6, 4) sits 6 m east and 4 m north of the origin. Negative values are simply west of or south of it.

In 3D: add Z for height

BIM tools (Revit, ArchiCAD, IFC files, AutoCAD, Rhino) add a third axis, Z, pointing up. So a point on level 2 at 3.5 m might be (6, 4, 3.5). Plans look straight down the Z axis; elevations look along X or Y.

Turn the axes below, move the point and switch between conventions. The dashed lines walk from the origin to the point: across the floor first, then straight up. Notice which letter “up” has in each convention.

Right-handed and left-handed

Once X and Y are fixed there are two choices for the third axis. In a right-handed system, point the fingers of your right hand along +X, curl them towards +Y, and your thumb points along +Z. A left-handed system uses the left hand, so its Z points the other way.

• Handedness also fixes which way is a positive rotation: in a right-handed system, a positive angle about Z turns counter-clockwise when you look down from above. That is the convention used throughout this course.
• It decides the direction of the cross product (coming in the Vectors module), which is how software finds a face’s outward normal.
• Mixing the two silently mirrors a model: left becomes right, and every door swings the wrong way.

Converting Y-up to Z-up

Exporting a model to a web viewer or a game engine, or importing a glTF file, means converting between conventions. For two right-handed systems the rule is short: the Y-up “up” becomes Z, and the Y-up “towards the viewer” axis becomes minus Y.

1using P3 = (double X, double Y, double Z);2 3static P3 YUpToZUp(P3 p) => (p.X, -p.Z, p.Y);4static P3 ZUpToYUp(P3 p) => (p.X, p.Z, -p.Y);5 6P3 columnTop = (4, 3, -2);          // from a glTF file: Y is up7P3 inModel = YUpToZUp(columnTop);8Console.WriteLine($"Z-up: {inModel}");9Console.WriteLine($"Back: {ZUpToYUp(inModel)}");

Z-up: (4, 2, 3)
Back: (4, 3, -2)

Screens point Y down

Screens, images, SVG and WPF all put the origin at the top-left with Y growing downwards. To draw a plan you flip Y and scale from metres to pixels. Every widget on this page does exactly this.

1// Plan (metres, Y up) to a canvas (pixels, Y down) that is 'height' pixels tall.2static (double X, double Y) ToScreen(double x, double y, double scale, double height)3    => (x * scale, height - y * scale);4 5Console.WriteLine(ToScreen(0, 0, 50, 400));     // origin: bottom-left corner6Console.WriteLine(ToScreen(6, 4, 50, 400));

(0, 400)
(300, 200)

Several origins in one project

A real project has more than one coordinate system. The internal origin is where the software measures from; a project base point gives the building a convenient origin (often a grid intersection); the survey point ties the model to real-world site coordinates. Elements inside a family or block have their own local coordinates too. The Matrices module shows how to move between all of them.

1using P3 = (double X, double Y, double Z);2 3static P3 YUpToZUp(P3 p) => (p.X, -p.Z, p.Y);4 5P3[] fromFile = { (1, 3.5, -2), (4, 7.2, -1), (2, 6, -5) };6 7P3 highest = YUpToZUp(fromFile[0]);8foreach (P3 p in fromFile)9{10    P3 q = YUpToZUp(p);11    if (q.Z > highest.Z) highest = q;12}13Console.WriteLine($"Highest: {highest}");

Highest: (4, 1, 7.2)