A3DGE - Amateur 3D Game EngineIntroduction Units Class Hierarchy Classes, Interfaces, Objects and Records Types Variables Constants Functions and Procedures Identifiers Conclusion
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Unit a3dge.Math3D
Uses Classes, Interfaces, Objects and Records Variables
Description
Some useful mathematic functions.
This unit is mostly for internal use but it is also useful in some cases; for example if you're building or modifying your own 3D geometry.
To use these structs and routines you should have some knowledge of Euclidean Geometry and Linear algebra, matrices, vectors and such.
Remember: OpenGL uses degrees, not radians.
Overview
Functions and Procedures
function Vector3D (const aX, aY, aZ: GLfloat): TVector3D; inline; |
function Vector4D (const aX, aY, aZ, aW: GLfloat): TVector4D; inline; |
function AreSameVector3D (const aV1, aV2: TVector3D): Boolean; inline; |
function AreDifferentVector3D (const aV1, aV2: TVector3D): Boolean; inline; |
FUNCTION GetVectorLength (CONST Vector: TVector3D): SINGLE; INLINE; |
FUNCTION GetVectorLength2 (CONST Vector: TVector3D): SINGLE; INLINE; |
FUNCTION GetNormalizedVector (CONST Vector: TVector3D): TVector3D; |
function GetOppositeVector (const aVector: TVector3D): TVector3D; |
FUNCTION AddVectors (CONST V1, V2: TVector3D): TVector3D; INLINE; |
FUNCTION SubVectors (CONST V1, V2: TVector3D): TVector3D; INLINE; |
function MultiplyVector (const aVector: TVector3D; const aValue: GLfloat) : TVector3D; inline; |
function DivideVector (const aVector: TVector3D; const aValue: GLfloat) : TVector3D; inline; |
FUNCTION VectorAsString (CONST Vector: TVector3D): STRING; |
FUNCTION GetCrossProduct (CONST aV1, aV2: TVector3D): TVector3D; |
FUNCTION GetDotProduct (CONST aV1, aV2: TVector3D): SINGLE; |
function GetVectorLerp (const V1, V2: TVector3D; const T: Real): TVector3D; |
FUNCTION GetPlaneNormal (CONST v1, v2, v3: TVector3D): TVector3D; |
FUNCTION GetDistance (CONST P1, P2: TVector3D): SINGLE; |
FUNCTION GetDistance2 (CONST P1, P2: TVector3D): SINGLE; |
FUNCTION GetDistance2Segment (CONST P, A, B: TVector3D): SINGLE; |
FUNCTION GetDistance2Segment2 (CONST P, A, B: TVector3D): SINGLE; |
PROCEDURE ScaleVectorList ( var VectorList: TVector3DList; CONST Scale: SINGLE ); OVERLOAD; |
PROCEDURE ScaleVectorList ( var VectorList: TVector3DList; CONST Scale: TVector3D ); OVERLOAD; |
function IdentityMatrix: TMatrix3D; |
function GetTranslationMatrix (const x, y, z: GLfloat): TMatrix3D; overload; |
function GetTranslationMatrix (const V: TVector3D): TMatrix3D; overload; |
function GetScalingMatrix (const x, y, z: GLfloat): TMatrix3D; overload; |
function GetScalingMatrix (const S: TVector3D): TMatrix3D; overload; |
function GetXRotateMatrix (const r: GLfloat): TMatrix3D; |
function GetYRotateMatrix (const r: GLfloat): TMatrix3D; |
function GetZRotateMatrix (const r: GLfloat): TMatrix3D; |
function GetRotationMatrix (const x, y, z: GLfloat): TMatrix3D; overload; |
function GetRotationMatrix (const R: TVector3D): TMatrix3D; overload; |
function GetTransformationMatrix ( const aScale, aXrot, aYrot, aZrot, aXtrans, aYtrans, aZtrans: GLfloat ): TMatrix3D; overload; |
function GetTransformationMatrix ( const aScale: GLfloat; aRot, aTrans: TVector3D ): TMatrix3D; overload; |
function GetTransformationMatrix ( const aScaleX, aScaleY, aScaleZ, aXrot, aYrot, aZrot, aXtrans, aYtrans, aZtrans: GLfloat ): TMatrix3D; overload; |
function GetTransformationMatrix ( const aScale, aRot, aTrans: TVector3D ): TMatrix3D; overload; |
function MultiplyMatrix (const A, B: TMatrix3D): TMatrix3D; |
function MultiplyVectorMatrix ( const aM: TMatrix3D; const aV: TVector3D ): TVector3D; inline; |
function RotateVector (const aM: TMatrix3D; const aV: TVector3D): TVector3D; inline; |
procedure SetGLMatrix (const aM: TMatrix3D); |
FUNCTION GetRelativeDirection ( CONST Chaser, Target: TVector3D; CONST ChaserAngle: SINGLE ): INTEGER; INLINE; |
FUNCTION GetBoundingSphereRadius (CONST VectorList: TVector3DList): SINGLE; |
Types
Constants
Description
Functions and Procedures
function Vector3D (const aX, aY, aZ: GLfloat): TVector3D; inline; |
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Build a 3D vector.
See also
- Vector4D
- Build a 4D vector.
- GetOppositeVector
- Calculate the opposite vector.
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function Vector4D (const aX, aY, aZ, aW: GLfloat): TVector4D; inline; |
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Build a 4D vector.
See also
- Vector3D
- Build a 3D vector.
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function AreSameVector3D (const aV1, aV2: TVector3D): Boolean; inline; |
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Check if vectors are the same.
See also
- AreDifferentVector3D
- Check if vectors are different.
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function AreDifferentVector3D (const aV1, aV2: TVector3D): Boolean; inline; |
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Check if vectors are different.
See also
- AreSameVector3D
- Check if vectors are the same.
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FUNCTION GetVectorLength (CONST Vector: TVector3D): SINGLE; INLINE; |
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Calculate length of the vector.
If you only need to compare lengths of vectors, you can use GetVectorLength2 wich is faster.
Parameters
- Vector
- A vector.
Returns
Length Vector. See also
- GetVectorLength2
- Calculate the squared length of the vector.
- GetNormalizedVector
- Return the given vector with a magitude (length) of
1.
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FUNCTION GetVectorLength2 (CONST Vector: TVector3D): SINGLE; INLINE; |
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Calculate the squared length of the vector.
To calculate vector length a square root is used, wich is slow. Calculating the squared length is quite faster as it only uses addition and multiplication. Since squares are proportional to the base, compare squared lengths have the same result than compare lengths but faster.
Parameters
- Vector
- A vector.
Returns
Squared length Vector. See also
- GetVectorLength
- Calculate length of the vector.
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FUNCTION GetNormalizedVector (CONST Vector: TVector3D): TVector3D; |
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Return the given vector with a magitude (length) of 1.
If vector is too small then vector will became a zero vector (i.e. a vector with all components equal zero).
Parameters
- Vector
- A vector.
Returns
A vector in the same direction than Vector but length 1 (one). See also
- GetVectorLength
- Calculate length of the vector.
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function GetOppositeVector (const aVector: TVector3D): TVector3D; |
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Calculate the opposite vector.
This returns a vector with same direction and magnitude but opposite direction.
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FUNCTION VectorAsString (CONST Vector: TVector3D): STRING; |
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Build a string with vector values.
Useful for logging and debugging.
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FUNCTION GetCrossProduct (CONST aV1, aV2: TVector3D): TVector3D; |
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Calculate the cross product between the given vectors.
Parameters
- aV1
- A vector. No need to be normalized.
- aV2
- A vector. No need to be normalized.
Returns
A vector that is perpendicular to both aV1 and aV2. See also
- GetDotProduct
- Calculates dot product of given vectors.
- GetNormalizedVector
- Return the given vector with a magitude (length) of
1.
- GetPlaneNormal
- Get the normal of the plane defined by the given vertices.
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FUNCTION GetDotProduct (CONST aV1, aV2: TVector3D): SINGLE; |
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Calculates dot product of given vectors.
For normalized vectors, the result is proportional to the angle between both vectors, returning 1 if both point in the same direction, 0 if they're perpendicular or -1 if they're in opposite directions.
See also
- GetCrossProduct
- Calculate the cross product between the given vectors.
- GetPlaneNormal
- Get the normal of the plane defined by the given vertices.
- GetNormalizedVector
- Return the given vector with a magitude (length) of
1.
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function GetVectorLerp (const V1, V2: TVector3D; const T: Real): TVector3D; |
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Calculate linear interpolation between given vectors.
This is most commonly used to find a point some fraction of the way along a line between two endpoints (e.g. to move an object gradually between those points).
Parameters
- V1
- Start value, returned when
T is 0.
- V2
- End value, returned when
T is 1.
- T
- Value used to interpolate between
V1 and V2.
Returns
Interpolated value, equals to ((V2 - V1) * T) + V1. |
FUNCTION GetPlaneNormal (CONST v1, v2, v3: TVector3D): TVector3D; |
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Get the normal of the plane defined by the given vertices.
The vectors passed as parameters are points in the plane. Such points should be passed in clockwise order.
Parameters
- v1
- A vector representing a point in a plane.
- v2
- A vector representing a point in a plane.
- v3
- A vector representing a point in a plane.
Returns
A vector that is perpendicular to the plane defined by the parameters. See also
- GetCrossProduct
- Calculate the cross product between the given vectors.
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FUNCTION GetDistance (CONST P1, P2: TVector3D): SINGLE; |
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Calculate the distace between two points.
See also
- GetDistance2
- Calculate the square distance between two points.
- GetDistance2Segment
- Calculate the distance from point
P to line segment A - B.
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FUNCTION GetDistance2 (CONST P1, P2: TVector3D): SINGLE; |
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Calculate the square distance between two points.
This function is faster than GetDistance, so compare square of distance is faster than compare distance.
This function can be used to check sphere collisions:
type
TSphere = record
Center: TVector3D;
Radius: Single
end;
function CheckSphereCollision (aSphere1, aSphere2: TSphere): Boolean;
begin
Result := GetDistance2 (aSphere1.Center, aSphere2.Center) <= Sqr (aSphere1.Radius + aSphere2.Radius)
end;
See also
- GetDistance
- Calculate the distace between two points.
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FUNCTION GetDistance2Segment (CONST P, A, B: TVector3D): SINGLE; |
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Calculate the distance from point P to line segment A - B.
See also
- GetDistance2Segment2
- Calculate the square of the distance from point
P to line segment A - B.
- GetDistance
- Calculate the distace between two points.
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FUNCTION GetDistance2Segment2 (CONST P, A, B: TVector3D): SINGLE; |
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Calculate the square of the distance from point P to line segment A - B.
This function is faster than GetDistance2Segment, so compare square of distance is faster than compare distance.
It can be used to check for collisions but it is slower than sphere or bounding-box collision algorithms.
See also
- GetDistance2Segment
- Calculate the distance from point
P to line segment A - B.
- GetDistance2
- Calculate the square distance between two points.
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PROCEDURE ScaleVectorList ( var VectorList: TVector3DList; CONST Scale: SINGLE ); OVERLOAD; |
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Scale all vertex of the list.
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PROCEDURE ScaleVectorList ( var VectorList: TVector3DList; CONST Scale: TVector3D ); OVERLOAD; |
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Scale all vertex of the list.
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function GetTranslationMatrix (const x, y, z: GLfloat): TMatrix3D; overload; |
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Construct a 3D translation matrix.
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function GetScalingMatrix (const x, y, z: GLfloat): TMatrix3D; overload; |
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Construct a 3D scaling matrix.
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function GetXRotateMatrix (const r: GLfloat): TMatrix3D; |
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Construct a 3D tranformation matrix which will rotate points around the x axis by the specified amount (given in degrees).
See also
- GetYRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the t axis by the specified amount (given in degrees).
- GetZRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the z axis by the specified amount (given in degrees).
- GetRotationMatrix
- Construct a 3D translation matrix that rotate points around all three axis by the specified amount (given in degrees).
- GetTransformationMatrix
- Construct a full 3D transformation matrix.
- MultiplyVectorMatrix
- Multiply the given vector with the given matrix.
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function GetYRotateMatrix (const r: GLfloat): TMatrix3D; |
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Construct a 3D tranformation matrix which will rotate points around the t axis by the specified amount (given in degrees).
See also
- GetXRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the x axis by the specified amount (given in degrees).
- GetZRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the z axis by the specified amount (given in degrees).
- GetRotationMatrix
- Construct a 3D translation matrix that rotate points around all three axis by the specified amount (given in degrees).
- GetTransformationMatrix
- Construct a full 3D transformation matrix.
- MultiplyVectorMatrix
- Multiply the given vector with the given matrix.
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function GetZRotateMatrix (const r: GLfloat): TMatrix3D; |
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Construct a 3D tranformation matrix which will rotate points around the z axis by the specified amount (given in degrees).
See also
- GetXRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the x axis by the specified amount (given in degrees).
- GetYRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the t axis by the specified amount (given in degrees).
- GetRotationMatrix
- Construct a 3D translation matrix that rotate points around all three axis by the specified amount (given in degrees).
- GetTransformationMatrix
- Construct a full 3D transformation matrix.
- MultiplyVectorMatrix
- Multiply the given vector with the given matrix.
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function GetRotationMatrix (const x, y, z: GLfloat): TMatrix3D; overload; |
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Construct a 3D translation matrix that rotate points around all three axis by the specified amount (given in degrees).
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function GetRotationMatrix (const R: TVector3D): TMatrix3D; overload; |
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Construct a 3D translation matrix that rotate points around all three axis by the specified amount (given in degrees).
See also
- GetXRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the x axis by the specified amount (given in degrees).
- GetYRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the t axis by the specified amount (given in degrees).
- GetZRotateMatrix
- Construct a 3D tranformation matrix which will rotate points around the z axis by the specified amount (given in degrees).
- GetScalingMatrix
- Construct a 3D scaling matrix.
- GetTranslationMatrix
- Construct a 3D translation matrix.
- GetTransformationMatrix
- Construct a full 3D transformation matrix.
- RotateVector
- The same than MultiplyVectorMatrix but assumes that matrix is rotation only.
- MultiplyVectorMatrix
- Multiply the given vector with the given matrix.
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function GetTransformationMatrix ( const aScale, aXrot, aYrot, aZrot, aXtrans, aYtrans, aZtrans: GLfloat ): TMatrix3D; overload; |
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Construct a full 3D transformation matrix.
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function GetTransformationMatrix ( const aScale: GLfloat; aRot, aTrans: TVector3D ): TMatrix3D; overload; |
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Construct a full 3D transformation matrix.
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function GetTransformationMatrix ( const aScaleX, aScaleY, aScaleZ, aXrot, aYrot, aZrot, aXtrans, aYtrans, aZtrans: GLfloat ): TMatrix3D; overload; |
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Construct a full 3D transformation matrix.
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procedure SetGLMatrix (const aM: TMatrix3D); |
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Replace current OpenGL matrix with the given one.
Note this will replace the current OpenGL matrix stack.
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FUNCTION GetRelativeDirection ( CONST Chaser, Target: TVector3D; CONST ChaserAngle: SINGLE ): INTEGER; INLINE; |
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Returns the direction where the Target relative to the Chaser.
Note this function works on the x/z plane only.
Parameters
- Chaser
- Position of the chaser object.
- Target
- Position of the target object.
- Direction
- The
x/z angle the Chaser is facing.
Returns
A value in the interval 0..7 where 0 is in front of, 1 is in the front/right, 2 is on the right and so one. |
FUNCTION GetBoundingSphereRadius (CONST VectorList: TVector3DList): SINGLE; |
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Calculate the radius of the sphere that encloses all points in the list.
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Types
TVector3DList = ARRAY OF TVector3D; |
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List of vectors.
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TVector4D = ARRAY [Vx..Vw] OF GLfloat; |
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A 4D vector.
A TVector4D can be used to represent a plane, where components Vx, Vy and Vz is the plane normal and Vw is the distance to the origin.
See also
- Vector4D
- Build a 4D vector.
- TVector3D
- A 3D vector.
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TMatrix3D = ARRAY [0..15] OF GLfloat; |
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A matrix in array form.
You can multiply two matrices together to produce a third matrix, and this will have the same effect on points as applying the original two matrices one after the other. For example, if you have one matrix that rotates a point and another that shifts it sideways, you can combine them to produce a matrix that will do the rotation and the shift in a single step. You can build up extremely complex transformations in this way, while only ever having to multiply each point by a single matrix.
This array is zero based and the order it is stored is the one OpenGL expects:
| 0 4 8 12 |
| 1 5 9 13 |
| 2 6 10 14 |
| 3 7 11 15 | <- Note A3DGE doesn't use this row!
ˆ----- Translation column.
So to acces to element M(m,n) use M[((n - 1) * 4) + (m - 1)].
A3DGE assumes row 4 values are { 0 0 0 1 } in any case so you can ignore such row.
See also
- IdentityMatrix
- Build the identity matrix.
- GetTransformationMatrix
- Construct a full 3D transformation matrix.
- TVector3D
- A 3D vector.
- MultiplyVectorMatrix
- Multiply the given vector with the given matrix.
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Constants
Deg2Rad = ALLEGRO_TAU / 360; |
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Value to convert degrees to radians.
Usage
Radians := Degrees * Deg2Rad;
See also
- Rad2Deg
- Value to convert radians to degrees.
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Rad2Deg = 360 / ALLEGRO_TAU; |
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Value to convert radians to degrees.
Usage
Degrees := Radians * Deg2Rad;
See also
- Deg2Rad
- Value to convert degrees to radians.
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Vx = 0; |
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Identifies the x component of a vector.
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Vy = 1; |
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Identifies the y component of a vector.
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Vz = 2; |
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Identifies the z component of a vector.
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Vw = 3; |
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Identifies the w component of a vector.
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Vector0: TVector3D = (0 ,0 ,0); |
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Vector zero. For example, the center of an space.
See also
- VectorFront
- Vector that points to the "front" from a center.
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VectorFront: TVector3D = (0, 0, -1); |
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Vector that points to the "front" from a center.
See also
- Vector0
- Vector zero.
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