Vectors have multiple uses in
3D graphics, from describing distance and direction to speed. Unlike a
point, which has only a position, a vector has both a direction and a
length (magnitude), allowing it to be utilized to determine which
direction a polygon is facing, which direction an object or particle is
heading or just describe its position. Typically, vectors are designated
as arrows with a head and tail showing the direction and magnitude. Figure 1 shows an example of a vector within a coordinate system.
Vectors typically fall into two categories, free vectors and fixed vectors. The vector shown in Figure 5.6 is an example of a free vector.
A free vector is a vector that can be placed in an arbitrary location within a coordinate system and its meaning doesn’t change.
A fixed vector remains fixed to the origin of the
coordinate system. This places the tail of these vectors at the origin
while the head is placed at a location in space. Because fixed vectors
are centered on the origin, it allows them to be used to designate a
position. This position is comprised of three scalar values called
components. The number of components within the vector corresponds to
the number of axes. For example, if the coordinate system is describing
3D space, then three components are needed to describe a position within
it. For example, the vector <3, 2, 7> corresponds to a position 3
units away from the origin on the X axis, 2 units away on the Y axis,
and 7 units away on the Z axis.
Direct3D has a few types defined in the D3DX library that are useful when working with vectors.
The D3DXVECTOR2 type is a structure containing two components, an X and a Y.
typedef struct D3DXVECTOR2 {
FLOAT x;
FLOAT y;
} D3DXVECTOR2;
The D3DXVECTOR3 type contains three components, an X, Y, and a Z, within its structure.
typedef struct D3DXVECTOR3 {
FLOAT x;
FLOAT y;
FLOAT z;
} D3DXVECTOR3;
While you could define these types yourself, the
Direct3D structures contain more than just the definitions of the
embedded components. The structures contain functions and operator
overloads relevant to the specific vector type, allowing the vectors to
be manipulated easily.
Note
If you want to see all the functionality provided
the vector types by D3DX, look at the D3DX10math.h header file in the
DirectX SDK.
When using vectors you’re bound to come across a few different types; here’s a small list:
Position Vector—
A type of vector that is used to describe a position within a
coordinate system, with the tail being at the origin and the head at a
point.
Normal Vector— A vector that is perpendicular to a plane. This is useful for determining whether a polygon is front or back facing.
Unit Vectors—
A vector that has a length of 1. Not all vectors require a large
length. When creating a directional light, only the direction of the
vector is important.
Zero Vectors— A vector with a length of 0.
Determining the Length of a Vector
Occasionally
it is useful to know the length of a vector—since the length or
magnitude of the vector can be used as acceleration or velocity when
applied to a game object. The length is also used as an input when
normalizing the vector.
To calculate the length of a vector, each of the
components must be first squared and then added together. Finally, the
square root of this number is taken to give the output length.
sqrt(vectorX * vectorX + vectorY * vectorY + vectorZ * vectorZ);
Direct3D provides a function called D3DXVec3Length that can be used to calculate a vector length.
FLOAT D3DXVec3Length( CONST D3DXVECTOR3 *pV );
Normalize a Vector
Normalizing a vector is the process of reducing
any length vector into a unit vector. This is best done when only the
direction of a vector is needed and the length is unimportant. Vectors
can be normalized simply by dividing each of the components by the
vector’s length.
vectorX = (vectorX / length);
vectorY = (vectorY / length;
vectorZ = (vectorZ / length);
The final vector will still point in the same direction, but the length of the vector is reduced to 1.
Direct3D offers a function to perform this for you called D3DXVec3Normalize.
D3DXVECTOR3 * D3DXVec3Normalize(
D3DXVECTOR3 *pOut,
CONST D3DXVECTOR3 *pV
);
This function takes two parameters. The first parameter is a pointer to the D3DXVECTOR3 object to be filled with the normalize vector. The last parameter is the original vector.
Cross Product
A
cross product of a vector is used to calculate a normal vector. A
normal vector or normal is used when performing lighting to determine
the orientation of a particular polygon. The polygon’s orientation is
used to figure out how much light the polygon is receiving.
Normals
can be calculated on a polygon or vertex basis and have vastly
different visual results. Because there are more vertices in an object
than polygons, calculating normals on a per-vertex basis is more
complicated but yields much better visual results.
Calculating the normal vector using the cross
product requires two vectors from an existing polygon. The resulting
vector will be perpendicular to the input vectors.
newVectorX = (vector1Y * vector2Z) - (vector1Z * vector2Y);
newVectorY = (vector1Z * vector2X) - (vector1X * vector2Z);
newVectorZ = (vector1X * vector2Y) - (vector1Y * vector2X);
Direct3D provides the D3DXVec3Cross function for calculating the cross product.
D3DXVECTOR3* D3DXVec3Cross( D3DXVECTOR3 *pOut,
CONST D3DXVECTOR3 *pV1,
CONST D3DXVECTOR3 *pV2 );
The D3DXVec3Cross function takes three
parameters. The first parameter is the output vector for the
calculation. The second and third parameters are the existing polygon
vectors.
Dot Product
The final vector calculation I’ll go over is the
dot product. The dot product is used to determine the angle between two
vectors and is commonly used for back-face culling. Back-face culling is
the process by which polygons that are not visible are removed to
reduce the number of polygons being drawn. If two vectors have an angle
less than 90 degrees, then the dot product is a positive value;
otherwise, the dot product is negative. The sign of the dot product is
what’s used to determine whether a polygon is front or back facing.
Polygons facing away from the viewer are not shown.
The
dot product is calculated based on two existing vectors of a polygon.
The components of the two vectors are multiplied and then added together
to create the dot product. Before calculating the dot product, always
normalize the two input vectors.
Float dotProduct = Vector1X * vector2X + vector1Y * vector2Y + vector1Z *
vector2Z;
Direct3D contains the function D3DXVec3Dot for calculating the dot product.
FLOAT D3DXVec3Dot( CONST D3DXVECTOR3 *pV1, CONST D3DXVECTOR3 *pV2 );
The D3DXVec3Dot function takes the two input vectors as parameters.
