unity-3d-math

3D Math & Spatial Reasoning -- Correctness Patterns

Prerequisite skills: unity-scripting (Vector3/Quaternion API), unity-foundations (Transform, GameObjects), unity-physics (raycasting basics)

These patterns target the most dangerous Claude failure mode: plausible-looking math that compiles but produces wrong results. Each pattern documents Claude's default mistake and the correct Unity approach.

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PATTERN: Coordinate Space -- TransformPoint vs TransformDirection

WHEN: Converting between local and world space

WRONG (Claude default):

// Using TransformDirection for a position offset
Vector3 worldPos = transform.TransformDirection(localOffset);

RIGHT:

// Positions: TransformPoint (applies position + rotation + scale)
Vector3 worldPos = transform.TransformPoint(localOffset);

// Directions: TransformDirection (applies rotation only, ignores scale)
Vector3 worldDir = transform.TransformDirection(localDir);

// Vectors: TransformVector (applies rotation + scale, no position)
Vector3 worldVec = transform.TransformVector(localVec);

GOTCHA: TransformDirection ignores scale -- if the parent has non-uniform scale and you need the direction scaled, use TransformVector. For the inverse operations, use InverseTransformPoint, InverseTransformDirection, InverseTransformVector.

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PATTERN: Quaternion Multiplication Order

WHEN: Combining rotations (e.g., applying a local rotation on top of a world rotation)

WRONG (Claude default):

// "First rotate by A, then rotate by B"
transform.rotation = rotA * rotB; // Actually applies B first, then A

RIGHT:

// Quaternion multiplication applies RIGHT operand first
// "Apply B in A's space" = A * B
// Parent-then-child: parent * child
transform.rotation = worldRotation * localRotation;

// Example: rotate 45 degrees around Y, then tilt 30 degrees around local X
Quaternion yaw = Quaternion.AngleAxis(45f, Vector3.up);
Quaternion pitch = Quaternion.AngleAxis(30f, Vector3.right);
transform.rotation = yaw * pitch; // yaw is applied in world, pitch in yaw's local space

GOTCHA: This is the opposite of matrix multiplication reading order. If you think "first A then B", write A * B -- the right operand is applied in the left operand's local space.

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PATTERN: Euler Angle Interpolation (Gimbal Lock)

WHEN: Smoothly rotating between two orientations

WRONG (Claude default):

// Interpolating euler angles directly
Vector3 currentEuler = Vector3.Lerp(startEuler, endEuler, t);
transform.eulerAngles = currentEuler;

RIGHT:

// Always interpolate quaternions, never euler angles
transform.rotation = Quaternion.Slerp(startRot, endRot, t);

// For small angles where performance matters, Lerp is acceptable
transform.rotation = Quaternion.Lerp(startRot, endRot, t); // Slightly faster, less accurate for large arcs

GOTCHA: Euler angles suffer from gimbal lock at 90-degree pitch and have discontinuities (e.g., 359 to 1 degree jumps through 358 degrees instead of 2). Quaternion.Slerp always takes the shortest path. Use Quaternion.Lerp only when the angular difference is small (< 45 degrees).

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PATTERN: Quaternion.LookRotation Zero Vector

WHEN: Rotating an object to face a target direction

WRONG (Claude default):

Vector3 dir = target.position - transform.position;
transform.rotation = Quaternion.LookRotation(dir);

RIGHT:

Vector3 dir = target.position - transform.position;
if (dir.sqrMagnitude > 0.0001f) // Guard against zero/near-zero vector
{
    transform.rotation = Quaternion.LookRotation(dir, Vector3.up);
}

GOTCHA: LookRotation(Vector3.zero) produces NaN quaternion that silently corrupts the transform. The forward parameter does not need to be normalized (Unity normalizes internally), but it MUST be non-zero. Also fails if forward is exactly parallel to up -- the second parameter defaults to Vector3.up, which breaks if the target is directly above/below.

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PATTERN: Never Modify Quaternion Components Directly

WHEN: Trying to zero out or modify a specific rotation axis

WRONG (Claude default):

// "Remove the X rotation"
Quaternion rot = transform.rotation;
rot.x = 0f;
transform.rotation = rot;

RIGHT:

// Extract euler, modify, rebuild
Vector3 euler = transform.eulerAngles;
euler.x = 0f;
transform.rotation = Quaternion.Euler(euler);

// Or use Quaternion factory methods
// Keep only Y rotation:
transform.rotation = Quaternion.Euler(0f, transform.eulerAngles.y, 0f);

GOTCHA: Quaternion x/y/z/w are NOT euler angles. They are components of a 4D unit quaternion. Setting .x = 0 produces a non-unit quaternion with undefined behavior. Always use factory methods: Quaternion.Euler(), Quaternion.AngleAxis(), Quaternion.LookRotation().

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PATTERN: Float Comparison

WHEN: Comparing positions, distances, or any floating-point values

WRONG (Claude default):

if (transform.position == targetPosition) { /* arrived */ }
if (distance == 0f) { /* overlapping */ }

RIGHT:

// For positions: use sqrMagnitude with epsilon
if ((transform.position - targetPosition).sqrMagnitude < 0.0001f) { /* arrived */ }

// For single floats: use Mathf.Approximately
if (Mathf.Approximately(distance, 0f)) { /* close enough */ }

// For custom tolerance:
const float epsilon = 0.01f;
if (Mathf.Abs(a - b) < epsilon) { /* within tolerance */ }

GOTCHA: Vector3 == Vector3 in Unity does use an approximate comparison internally (epsilon ~1e-5), but it is often too tight for gameplay logic. Use explicit thresholds matching your game's precision needs. Never use == with calculated floats that accumulated error.

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PATTERN: Vector3.Angle is Always Positive

WHEN: Determining turn direction (left vs right, clockwise vs counter-clockwise)

WRONG (Claude default):

float angle = Vector3.Angle(transform.forward, dirToTarget);
// angle is always 0-180, cannot tell left from right

RIGHT:

// SignedAngle returns -180 to +180 relative to the specified axis
float signedAngle = Vector3.SignedAngle(transform.forward, dirToTarget, Vector3.up);
// Positive = target is to the right, Negative = target is to the left (when axis is up)

GOTCHA: The sign depends on the axis parameter. With Vector3.up as axis: positive = clockwise when viewed from above. Choose the axis that matches your rotation plane. For 2D games using XY plane, use Vector3.forward as the axis.

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PATTERN: Cross Product Order and Handedness

WHEN: Computing normals, perpendicular vectors, or winding order

WRONG (Claude default):

// Assuming right-hand rule
Vector3 normal = Vector3.Cross(edge1, edge2);

RIGHT:

// Unity uses a LEFT-handed coordinate system (Y-up, Z-forward)
// Cross product follows LEFT-hand rule:
// Cross(right, forward) = UP (not down)
Vector3 normal = Vector3.Cross(edge1, edge2);
// If normal points wrong way, swap operand order:
Vector3 flippedNormal = Vector3.Cross(edge2, edge1);

GOTCHA: Unity is left-handed (Y-up, X-right, Z-forward). OpenGL/Blender are right-handed. If you're porting math from a right-handed reference, you need to flip the cross product order OR negate one axis. Triangle winding is clockwise = front-facing in Unity.

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PATTERN: sqrMagnitude for Distance Comparisons

WHEN: Comparing distances in performance-sensitive code (Update loops, many objects)

WRONG (Claude default):

// Vector3.Distance computes a square root every call
if (Vector3.Distance(a, b) < detectionRange)
{
    // detected
}

RIGHT:

// Compare squared distances -- avoids sqrt
float sqrRange = detectionRange * detectionRange;
if ((a - b).sqrMagnitude < sqrRange)
{
    // detected
}

GOTCHA: Cache sqrRange outside the loop -- do not recompute range * range per iteration. This optimization matters when checking N objects per frame (O(N) sqrt calls). For single checks, Vector3.Distance is perfectly fine -- do not micro-optimize one-off calls.

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PATTERN: Camera.ScreenToWorldPoint Z Depth

WHEN: Converting a screen position (mouse, touch) to a world position

WRONG (Claude default):

// z=0 gives a point ON the camera's near plane, not in the scene
Vector3 worldPos = Camera.main.ScreenToWorldPoint(Input.mousePosition);

RIGHT:

// Set z to the desired distance from the camera
Vector3 screenPos = Input.mousePosition;
screenPos.z = desiredDistance; // Distance from camera along its forward axis
Vector3 worldPos = Camera.main.ScreenToWorldPoint(screenPos);

GOTCHA: The z component of the input vector is the distance from the camera in world units along the camera's forward direction. For perspective cameras, z=0 returns a point at the camera's position. For orthographic cameras, z doesn't affect x/y but still sets depth. To place objects on a ground plane, use Physics.Raycast with Camera.ScreenPointToRay instead.

---

PATTERN: Camera.ViewportToWorldPoint

WHEN: Placing objects at viewport edges (HUD bounds, screen limits)

WRONG (Claude default):

// Missing z depth -- returns a point at the camera
Vector3 topRight = Camera.main.ViewportToWorldPoint(new Vector3(1f, 1f, 0f));

RIGHT:

// z = distance from camera where you want the world point
float distFromCamera = 10f;
Vector3 topRight = Camera.main.ViewportToWorldPoint(new Vector3(1f, 1f, distFromCamera));
Vector3 bottomLeft = Camera.main.ViewportToWorldPoint(new Vector3(0f, 0f, distFromCamera));

GOTCHA: Viewport coordinates are normalized: (0,0) = bottom-left, (1,1) = top-right. The z value is NOT a Z world coordinate -- it is the distance from the camera. This matters for perspective cameras where the frustum widens with distance.

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PATTERN: Plane.Raycast Semantics

WHEN: Finding where a ray intersects a mathematical plane

WRONG (Claude default):

Plane groundPlane = new Plane(Vector3.up, 0f);
float enter;
groundPlane.Raycast(ray, out enter);
Vector3 hitPoint = ray.GetPoint(enter); // Using enter without checking return value

RIGHT:

Plane groundPlane = new Plane(Vector3.up, 0f); // Normal=up, distance=0 (XZ plane at origin)
float enter;
if (groundPlane.Raycast(ray, out enter))
{
    Vector3 hitPoint = ray.GetPoint(enter);
}
// If returns false, the ray points away from the plane (enter is negative)

GOTCHA: Plane.Raycast returns true only when the ray intersects the plane's front side (the side the normal points toward). If the ray origin is behind the plane or pointing away, it returns false and enter is negative. The Plane constructor new Plane(normal, distance) -- the distance is the signed distance from origin along the normal. new Plane(Vector3.up, 5f) creates a plane at y = -5, NOT y = 5. Use new Plane(Vector3.up, new Vector3(0, 5, 0)) for a plane at y = 5.

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PATTERN: Bounds is Always Axis-Aligned (AABB)

WHEN: Checking spatial overlap or containment of rotated objects

WRONG (Claude default):

// Assuming bounds rotates with the object
if (renderer.bounds.Contains(point))
{
    // This is an AABB check, not an OBB check
}

RIGHT:

// renderer.bounds is an AXIS-ALIGNED bounding box in WORLD space
// It expands to encompass the rotated mesh, making it larger than the actual object
Bounds aabb = renderer.bounds;

// For oriented checks, use the collider or manual OBB:
// Option 1: Use a collider (accurate to shape)
Collider col = GetComponent<Collider>();
Vector3 closest = col.ClosestPoint(point);
bool inside = (closest - point).sqrMagnitude < 0.0001f;

// Option 2: Transform point to local space for local-space AABB check
Vector3 localPoint = transform.InverseTransformPoint(point);
Bounds localBounds = meshFilter.sharedMesh.bounds; // Local-space bounds
bool containsLocal = localBounds.Contains(localPoint);

GOTCHA: Renderer.bounds returns world-space AABB. Rotating an object makes its AABB grow (a rotated cube's AABB is larger than the cube). Mesh.bounds is local-space. For Bounds.Intersects(other), both must be in the same space. Collider.bounds is also AABB -- for precise shape checks use Physics.ComputePenetration or Collider.ClosestPoint.

---

PATTERN: Transform Hierarchy Scale Inheritance

WHEN: Working with parent/child transforms that have non-uniform scale

WRONG (Claude default):

// Assuming scale is independent
transform.localScale = Vector3.one; // Visually one-to-one? Not if parent is scaled

RIGHT:

// lossyScale gives the approximate world-space scale (read-only)
Vector3 worldScale = transform.lossyScale;

// To set a specific world-space scale on a child:
Vector3 parentScale = transform.parent.lossyScale;
transform.localScale = new Vector3(
    desiredWorldScale.x / parentScale.x,
    desiredWorldScale.y / parentScale.y,
    desiredWorldScale.z / parentScale.z
);

GOTCHA: lossyScale is only approximate when the hierarchy contains rotation + non-uniform scale (skew). Unity does not support skew in transforms, so lossyScale may not perfectly represent the visual scale. Avoid non-uniform scale on parents of rotated children. SetParent(parent, true) preserves world position/rotation/scale; SetParent(parent, false) keeps local values -- choose intentionally.

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Common Patterns Quick Reference

Need	Method	Space
Position local-to-world	TransformPoint	local -> world
Position world-to-local	InverseTransformPoint	world -> local
Direction local-to-world	TransformDirection	rotation only
Direction world-to-local	InverseTransformDirection	rotation only
Vector local-to-world	TransformVector	rotation + scale
Angle between vectors	Vector3.Angle	unsigned 0-180
Signed angle	Vector3.SignedAngle	-180 to +180
Dot product > 0	vectors face same hemisphere
Dot product == 0	vectors are perpendicular
Dot product < 0	vectors face opposite directions
Mouse to world ray	Camera.ScreenPointToRay	screen -> world
Mouse to world point	ScreenToWorldPoint + z depth	screen -> world
Viewport edge to world	ViewportToWorldPoint + z depth	viewport -> world

Related Skills

• unity-scripting -- Vector3, Quaternion, Time API reference
• unity-foundations -- Transform API, parent/child hierarchies
• unity-physics -- Raycasting, collider spatial queries
• unity-physics-queries -- Physics query correctness patterns

Additional Resources

• Vector3 API
• Quaternion API
• Transform API
• Camera API
• Plane API
• Bounds API