Geometry Classes in 2D and 3D in Java

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Geometry Classes in 2D and 3D
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Concept 2D Class Remarks 3D Class Remarks
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2 dimensions; has vector methods Vector3D 4 dimensions Matrix3D 4 4, access to all elements
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There are more differences between the Matrix3D class and the Matrix class than just their dimension. The Matrix3D class, because it exists only in Flash Player 10 and later, uses other features that are also only available in Flash Player 10, notably Vectors. And the Matrix3D class provides some extremely helpful utilities to ease working in three dimensions. Even further utilities are included in the Utils3D class, which is covered in 40, Advanced 3D.
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In actuality, the Vector3D class is a four-dimensional vector of the form <x, y, z, w>. You can use Vector3D to represent vectors and points in 3D by ignoring the w coordinate, leaving it set to 1. The earlier discussion on homogeneous coordinate space explains the purpose of using a 4D vector to represent 3D space. You can use Vector3D all day without ever thinking about the fourth coordinate. Unless you dig really, really deep, Flash Player handles it for you. This 4D vector can be used for purposes other than points and vectors in 3D space. It can also hold a row or column of a 4 4 matrix, represent a quaternion, or store the channels of a color as in <r, g, b, a>. A Vector3D stores each coordinate in the properties x, y, z, and w. These properties can also be set at construction time. The constructor takes these parameters, all optionally:
function Vector3D(x:Number = 0, y:Number = 0, z:Number = 0, w:Number = 0)
You can leave w off when you re representing 3D points. Here you create a new vector for the point (5, 5, 5):
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var v:Vector3D = new Vector3D(5, 5, 5);
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The Vector3D class has some static constants that represent the three axes:
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trace(Vector3D.X_AXIS); //Vector3D(1, 0, 0) trace(Vector3D.Y_AXIS); //Vector3D(0, 1, 0) trace(Vector3D.Z_AXIS); //Vector3D(0, 0, 1)
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Recall that a vector encodes both a direction and a magnitude. You can easily access the magnitude of a vector through its length and lengthSquared properties, as Example 34-4 shows. These properties are read-only. In math notation, you write the magnitude of a vector v like |v|. These vertical bars, although I have typeset them the same, are not the same bars that are used to outline a matrix. In other words, |v| is not a 1 1 matrix but the notation for the magnitude of v. EXAMPLE 34-4
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var v:Vector3D = new Vector3D(8, 4, 1); trace(v.length); //9 trace(v.lengthSquared); //81
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Of course, the magnitude of a vector is its length, so you can calculate its magnitude with the Cartesian distance formula x2 + y2 + z2 . All the basic vector arithmetic you can do is readily performed with the Vector3D class.
add(a:Vector3D):Vector3D, subtract(a:Vector3D):Vector3D Performs vector addition and subtraction, returning the result of the sum. Vector addition sums each component individually:
<x1 , y1 > + <x2 , y2 > = <x1 + x2 , y1 + y2 >
incrementBy(a:Vector3D):void, decrementBy(a:Vector3D):void Performs
vector addition and subtraction destructively, replacing the contents of the vector with the result of the sum.
34: Geometric and Color Transformations
scaleBy(s:Number):void Performs scalar multiplication destructively, replacing the
contents of the vector with the result of the scalar multiplication. Scalar multiplication multiplies each component of the vector with the scalar:
k <x, y> = <kx, ky>
negate():void Negates a vector by multiplying it by 1. The operation is destructive,
replacing the vector s contents with its negation.
equals(toCompare:Vector3D, allFour:Boolean = false):Boolean Compares two vectors, returning true if the vectors are equal. By default, this method compares the vectors as if they have three dimensions, ignoring any differences in w. By passing true to the allFour parameter, the method compares all four dimensions of the vectors. Vector equality is
the equality of all the vectors components:
<x1 , y1 > = <x2 , y2 > iff x1 = x2 and y1 = y2
nearEquals(toCompare:Vector3D, tolerance:Number, allFour:Boolean = false):Boolean Compares two vectors, allowing for a certain amount of error. If the two vectors components differ by less than the tolerance value, they are considered equal.
In 7, Numbers, Math, and Dates, you learned that oating-point numbers are not in nitely precise. Using this method, you can ignore small rounding errors and still determine if the vectors are meant to be equal, or you can provide a higher tolerance and determine if the vectors are nearly equal. The allFour parameter acts as it does in the equals() method. As I m sure you noticed, some of the preceding operations are destructive and some are nondestructive. You can make copies of a Vector3D by calling its clone() method. The Vector3D class also provides vector operations like dot product and cross product. The dot product of two vectors is useful for projecting one vector onto another vector. In Figure 34-6, vector v is projected onto the x-axis. The length of the projection is given by the vector s dot product with a unit vector in the x direction. In mathematical notation, dot products are written with a dot: a b.