diff --git a/src/bin/flat/main.rs b/src/bin/flat/main.rs
index e887958..23596a9 100644
--- a/src/bin/flat/main.rs
+++ b/src/bin/flat/main.rs
@@ -148,15 +148,10 @@ fn rel_to_abs(space: &impl Metric, base: &Location, rel: Vec2, steps: usize) ->
 }
 
 /// Converts a position and a rotation to a [Location]. Only the X direction is preserved from `rot` to ensure the resulting Location describes an orthonormal coordinate system.
-///
-/// Mathematically, returned `rot` is `M^-1 * orthonormalize(M * rot)` where `M^2` is the metric tensor.
 fn put_object(space: &impl Metric, pos: Vec2, rot: Mat2) -> Location {
-	let x_dir = rot.x_axis;
 	let metric_sqrt = Mat2::from(space.sqrt_at(pos));
 	let metric_inv_sqrt = Mat2::from(space.sqrt_at(pos).inverse());
-	let fx = (metric_sqrt * x_dir).normalize();
-	let fy = vec2(-fx.y, fx.x); // TODO: respect orientation of `rot`
-	let rot = metric_inv_sqrt * mat2(fx, fy);
+	let rot = metric_inv_sqrt * (metric_sqrt * rot).orthonormalize();
 	Location { pos, rot }
 }