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1Scalar, Points and Vectors:
The basic geometric objects and relationship among them can be described using the three fundamental types called scalars, points and vectors.
1.1 Geometric Objects.
One of the fundamental geometric objects is a point. In 3D geometric system, point is a location in space. Point possesses only the location property, mathematically point neither a size nor a shape. Points are useful in specifying objects but not sufficient.
Scalars: Scalars are objects that obey a set of rules that are abstraction of the operations of ordinary arithmetic. Thus, addition and multiplication are defined and obey the usual rules such as commutativity and associativity and also every scalar has multiplicative and additive inverses.
Vector: Another basic object which has both direction and magnitude, however, vector does not have a fixed location in space.
Directed line segment shown in figure below connects two points has both direction i.e, orientation and magnitude i.e., its length so it is called as a vector
Q
P
because of vectors does not have fixed position, the directed line segments shown in figure below are identical because they have the same direction and magnitude.
Vector lengths can be altered by the scalar components, so the line segment A shown in figure below is twice the length of line segment B
B A=2B
We can also combine directed line segments as shown in figure below by using the head and tail rule
D=A+B B
A
We obtained new vector D from two vectors A and B by connecting head of A to tail of B. Magnitude and direction of vector D is determined from the tail of A to the head of B, we can call D has sum of A and B, so we can write it as D=A+B.
Consider the two directed line segments A and E shown in figure below with the same length but opposite direction. We can define the vector E in terms of A as E =-A, so the vector E is called inverse vector of A. The sum of vectors A and E is called Zero vector, which is denoted as 0, that has a zero magnitude and orientation is undefined.
A
E
1.2 Coordinate-Free Geometry
When we learned simple geometry, most of us started with a Cartesian approach, i.e., Points were at locations in space p=(x,y,z) but this approach was nonphysical because, points exist regardless of the location of anarbitrary coordinate system.Most geometric objects results are independent of the coordinate system; Example in Euclidean geometry, two triangles are identical if twocorresponding sides and the angle between them are identical.
1.3 The Mathematical View: vector and Affine spaces.
Scalars, points and vectors can be viewed as members of sets; then we can use different space for representing and manipulating these sets. One of the important mathematical space is Vector Space.
Vector Space contains two entities called Vectors and Scalars. In Vector space, Scalar-Vector Multiplication and Vector-Vector addition can be performed .Example for mathematical space is geometric operations on directed line segments.
Euclidian Space is an extension of a vector space that adds a measure of size or distance. Example Length of a line segment.
Affine Space is an extension of a vector space that includes additional type of object called point. Affine allows performingvector- point addition to get new point, Point-Point subtraction to getvector, vector-vector addition to get the new vector, Scalar-vector multiplication and scalar-scalar operations
1.4 The Computer Science View
We have seen scalars, points and vectors can be viewed as members of sets we can also view these as abstract data types (ADT’s).An ADT is a set of operations on data; the operations are defined independently of how data are represented internally or of how the operations are implemented. From a computational point of view, we should be able to declare geometric objects through code h
vector u, v;
point p, q;
scalar a, b;
using language features we can write the code to perform the operations on these geometric objects.
1.5 Geometric ADT’s
In this section we will see, how we can use scalars, vectors and points to form geometrical objects and to perform geometric operations. Here we will use Greek letters α, β ,γ, …. to denote scalars, uppercase letters P, Q, R, …. to denote points and lower case letters like u,v,w,…. to denote vectors.
The magnitude of a vector v is a real number denoted by |v|. The operation of vector – scalar multiplication has the property that, |αv|=|α||v| and the direction of αv is same as the direction of v if α is positive else the direction is reverse.
The subtraction of two points P and Q yields a vector v denoted by v = P-Q and the point vector addition yields P=P+v.
Given any three points P,Q,R, then (P - Q) + (Q – R) = P –R.
1.6 Lines
The sum of point and vector leads to the line in affine space. Consider all points of the form
P(α) = P0 + α d .Where P0 is an arbitrary point, d is an arbitrary vector and α is an scalar that vary over some range of values. Thisform is called parametric form of the line because points generated by varying the α lies on the line as shown in figure below. For α=0, the line passes through the point P0 and α is increased, all the points generated lie on the line in the direction of the vector d.
If α >= 0, then P(α) is the ray leaving Q in the direction v, If we use two points to define v, then
P(α) = Q + α (R-Q)=Q+ α v= α R + (1- α)Q For 0≤ α ≤ 1 we get all the points on the line segmentjoining R and Q
1.7 Convexity
An object is convex iff for any two points in the object all points on the line segment between these points are also in the object
Convex Not Convex
Consider the object defined by n points P1,P2,……….Pn,
The affine sum is P=α1P1+α2P2+…….αn.Pn.
it is defined if and only if α1+α2+…….+αn=1
The set of points formed by the affine sum of n points, under the additional restriction αi≥0, i=1,……n, is convex hull of the set of points shown in Figure below formed by stretching a tight-fitting surface over a given set A points-shrink wrapping the points.
1.8 Dot and Cross Products
Orientation between two vectors are defined by Dot(inner) and cross(outer) products of a vectors. The dot product of u&v is written as u,v. If u.v=0 then two vectors are orthogonal. The square of the magnitude of a vector is given |u|²=u.v The cosine of the angle between two vector is given by
cosθ = u.v / |u||v|
and |u| cosθ = u.v / |v| is the length of the orthogonal projection of u onto v, as shown in Figure below.
ө v
|u| cos θ
Thus the dot product express the geometric result that the shortest distance from point to the line segment v is obtained by drawing the vector orthogonal to v from the end of u.
A vector space has a dimension, which is the maximum number of linearly independent vectors that we can find. We can use dot product to construct 3 vectors which are orthogonal to each other. We can also use two non parallel vectors, u and v to determine third vector n that is orthogonal to them. This vector is the cross product. i.e., n = u * v
u*v
v
u
We can use the cross product to derive three mutually orthogonal in a 3D space from any two non parallel vectors.
Starting with u and v we compute n i.e. n = u * v, then compute w by w = u * n, so u, w & n are mutually orthogonal.
The magnitude of the cross product gives the magnitude of the sine of the angle θ between u & v,
|sin θ| = | u*v | / |u| |v|
Note that u, v and n form right handed co ordinate system.
1.1.10 Planes
A plane can be defined by a point and two vectors or by three points (not on the same line)
If the plane is defined using two vector and a point then the equation is P(α, β)=R+ α u+ β v
If the plane is defined using three point then the equation isP(a,b)=R+a(Q-R)+b(P-Q) as shown in figure above.
The line segment that joins P and Q is the set of points of the form
S(α) = αP+(1- α)Q, 0 ≤ α ≤1
From arbitrary point on this line segment form the line segment from this point to R, using second parameter
T(β) = β P+(1- β)Q, 0 ≤ β ≤1
Combining these two equations, we obtain the equation for the plane
T(α , β) = β[α P+(1- α)Q]+ (1 – β)R, 0 ≤ α, β ≤1
Every plane has a vector n normal (perpendicular, orthogonal) to it, From point-two vector form P(α, β)=R+ α u+ β v, we know we can use the cross product to find normal n = u v and the equivalent form (P(α)-P) n=0
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