Advance in Algebraic curves: Application to Computer Graphing

A B S T R A C T Differential geometry, a branch of mathematics that studies the geometry of curves, surfaces, and manifolds (the higher-dimensional analogs of surfaces). Curve is a smoothly flowing line (non sharp changes) or a curve must bend (change direction) but in Mathematics a straight line also a curve. We will present new definitions and theorems.


Introduction
We will present the racing of differential geometry and evolution since BC to reach such incredible development in our time and its relation to other scientific branches and its applications in the areas of life.It shows how the concept of geometry appeared in the works of both Riemann and Lobachevski [1] and [2]., which later proved the important of this geometry to many of life problems.At the end, we focus on the subject of the study of differential geometry because of its close link to other mathematic fields.
Definition, Postulates and Axioms: The geometric history back to the very early time, not only what we have of geometry facts, so that in this period directed to collect at the result together to be in logical order.Also the Greeks did a lot of work to develop geometry no thing appeared for us specially after Euclid's famous work appeared that named Euclid's famous elements.

Curve
We develop the mathematical tools needed to model and study a moving object.
The object might be moving in the plane [3] and [4].
Higher-order derivatives are defined analogously  Now we prove that every (a < x < b) is in I, if x is not upper bound or lower bound then, there exists two elements y and z in I such that y < x < z.So by the definition x in I, according to  and  belong to I we obtain the fourth types.Now, let I an interval has lower bounded but not upper bounded.Let a be the  of I, every element in I is ≥ a.We are going to prove that I contains all real numbers x > a, hence x is not a lower bounded and I contain y such that y < x by the same way, we have there exist z such that z > x hence y < x < z, so x belongs to I. According to  belongs to I or not we obtain two types of non-upper bounded intervals.Finally, if an interval I is not upper bounded or lower bounded and for every element x we can find two elements y and z in I such that y < x < z, that lead to x in I. Hence I = R.

Example2.8.
A logarithmic spiral means a plane curve of the form () = (exp () cos() , exp() sin()), and  ∈ ℝ, where ,  ∈ ℝ with  ≠ 0. It shows the restriction to [0, ∞) of a logarithmic spiral with  < 0. Use an improper integral to prove that such a restriction has finite arc length it makes infinitely many loops around the origin.

The inner product.
13.The angle between nonzero vectors x and y is defined as: Recall that x and y are called orthogonal if ⟨, ⟩ = 0 .They are called parallel if one of them is a scalar multiple of the other.
Definition 2.14.If ,  ∈ ℝ with || ≠ 0 , then there is a unique way to write x as a sum of two vectors: the first of which is parallel to y and the second of which is orthogonal to y.The vector ∥ is called the projection of X in the direction of Y .The signed length of  ⊥ (that is, the scalar λ ∈ ℝ such that Notation 2.17.A basis B of a vector space V over a field F is a linearly independent subset of V that spans V. Notation 2.18.The vector in the set  = { 1 , … ,   } are said to be linearly independent if the equation a1Y1 + a2Y2 + ... + anYn = 0, can only be satisfied by ai = 0 for i = 1, 2, ..., n.
Example 2.19.Prove that every orthonormal set in ℝ  must be linearly independent.
Then for any index 1 ≤ i ≤ k we have by orthonormal that Thus the set {V1, V2, V3} is orthogonal but not orthonormal.(1) If γ has constant norm (that is |γ()|, =  , for all t ∈ I), then  ′ () is orthogonal to γ()for all t ∈ I.

Acceleration.
Definition 2.25.Acceleration is a vector quantity that is defined as the rate at which an object changes its velocity.An object is accelerating if it is changing its velocity.Velocity is defined as a vector measurement of the rate and direction of motion or, in other terms, the rate and direction of the change in the position of an object.The following notational convention from physics: If ():  → ℝ  , is a regular curve.v(t) =  ′ () (the velocity function) Also we have by the physics interpretation of a(t) comes from the vector version of Newton's law: where m is the objects mass, and F (t) is the vector-valued force acting on the object at time t.

2-The arc length
() = ∫ () = ∫ ( 2 − 1, Example 2.44.For constant a, b, c > 0, consider the generalized helix define as Where is the curvature maximal and minimal? 1.6.Plane Curve.[A plane curve is any curve, which can be drawn on the plane.Some curves are fairly simple, like a circle, and will have fairly simple algebraic equations.Some are very complex, like your signature, and may be very difficult to describe with an equation.Plane curves were studied intensively from the seven-tenth through the nineteenth centuries.]In This section, I studied some specialized properties of regular plane curve (regular curves in the plan ℝ 2 ).Let as first con-sider the linear isomorphism [An isomorphism between two vector space V and W is a map f : V→ W such that: whose effect is to rotate the vector (x, y) by 90 degrees counterclockwise Definition 2.45.Let γ : I → ℝ 2 is a unit-speed plane curve.At any time t ∈ I.We call κs : I → ℝ the signed curvature function which define as 90 (()) , its negative if the curve is turning clockwise at t, and its positive if counterclockwise.

Space Curves.
In this short subsection we concern a smooth curve γ in the standard three dimensional Euclidean space E. Let this curve be defined (up to translations and rotations of E) by its curvature κ(s) and its torsion τ (s), the arguments is the arc-length parameter.( (2)  ′ = −||.

Rigid Motion.
A Rigid Motion (motions are the orientation preserving isometrics.) is the action of taking an object and moving it to a different location without altering its shape or size.For examples of a Rigid motion are translation and rotation .But reflection and glide reflection are isometrics, but are not motions.Where, Translation: It is a shifting of a shape, where all the shapes are moved in the same direction and the same distance.Shapes are simply translated in a direction without loss of orientation.
Reflection occurs when an image is flipped over along an axis.A way to en-visage this is by placing a small mirror along an object to act as an axis of reflection.Rotation To understand rotation, imagine sticking a pin through the duplicate copy of tracing paper and moving it around the pin, which serves as the center of rotation.
Glide reflection A glide reflection is a reflection around an axis, combined with a translation along the same axis.

ADDITIONAL TOPICS IN CURVE
This section studied more deeply into the geometry of curves, including some of the famous theorems in the field.The theory of curves is an old and extremely well developed mathematical topic.The interior is on ones left as one traverses γ; more precisely, for each t ∈ [a, b], R90(γ(t) ′ ) points toward the interior in the sense that there exists δ > 0 such that γ(t) + sR90(γ) ′ lies in the interior for all s ∈ (0, δ).Otherwise, γ is negatively oriented, in which case its

Lemma 2 .Proposition 2 . 49 . 1 2𝜋(
46.Let γ : I → ℝ 2 is a unit-speed plane curve.At any time t ∈ I With κs : I → ℝ then |κs(t)| = |κ(t)| Definition 2.47.If : I → ℝ 2 is a regular plane curve (not necessarily parameterized by arc length), then for all t ∈ I, If γ : I → ℝ 2 is a unit-speed plane, then there exists a smooth angle function, θ : I → ℝ, such that for all t ∈ I, we have v(t) = (cos θ(t), sin θ(t)).This function is unique up to adding an integer multiple of 2π.Definition 2.50.The rotation index of a unit-speed closed plane curve γ : [a, b] → ℝ 2 equals () − ()), where is the angle function from Proposition 3.5.The rotation index of a regular closed plane curve (not necessarily of unit speed) means the rotation index of an orientation preserving unit-speed re parameterization of it.

Proposition 3 . 5 . 1 2𝜋(Definition 3 . 6 .Definition 3 . 8 .
rotation index equals 1, and R90(γ) ' points toward the exterior for all t ∈ [a, b].If f : [a, b] → S ' is a continuous function with f (a) = f (b), then there exists a continuous angle function φ : [a, b] → ℝ such that for all t ∈ [a, b], we have f (t) = (cos φ(t), sin φ(t)).This function is unique up to adding an integer multiple of 2π.The degree of f is defined as the integer () − ().A piecewise-regular curve in R ' is a continuous function γ : [a, b] → ℝ  with a partition, a = t0 < t1 < ... < tn = b, such that the restriction, γi, of γ to each subinterval [ti, ti+1] is a regular curve.It is called closed if additionally γ(a) = γ(b), and simple if γ is one-toone on the domain [a, b).It is said to be of unit speed if each γi is of unit speed.Theorem 3.7.(Generalized Hopf s Umlaufsatz) Let : [a, b]→ ℝ 2 be a unit-speed positively oriented piecewise-regular simple closed plane curve.Let s denote its signed curvature function, and let i be the list of signed angles at its corners.Then Let γ : [a, b] → ℝ 2 be a piecewise-smooth simple closed plane curve with signed angles denoted by ℵ i.The ith interior angle of γ, denoted by βi ∈ [0, 2π], is defined as   = {  −  if  is positively oriented  +  if  is negatively oriented In theorem 3.7, γ is assumed to be positively oriented, so the theorem becomes ( − 2), where n is the number of corners.If the smooth segments of γ are straight-line segments, then this becomes ∑    = ( − 2).
Definition 2.4.Let :  → ℝ  be a curve.It is called regular if its speed is always nonzero (|ψ(t)| = 0 for all t ∈ I).It is called unit-speed or parameterized by arc length if its speed is always equal to 1 (|ψ(t)| = 0 for all t ∈ I).
, where t changes from t to t + h and ψ changes fron () to ( + ℎ) ℎ→0 ( + ℎ) − () ℎ Definition 2.6.A set I of ℝ is an interval if I contains two real numbers, so it's contains all the numbers between them.
To date.I have studied in this section the curvature and its properties and I read some theorems that connect between the unit tangent, unit normal and the curvature through the rate of change of velocity (Acceleration).
Definition 2.31.Suppose that : I → ℝ  is a regular curve.A reparametrization of  is a function of the form  ̃=  o φ → ℝ  , where I ˜ is an interval and φ : I ˜ → I is a smooth bijection with nowhere-vanishing derivative (φ ' (t) 0 for all t ∈ I ˜).And  ̃ is called orientation-preserving if φ ˜ > 0, and orientation-reversing if φ ˜ < 0. Proposition 2.32.A regular curve :  → ℝ  can be reparametrized by arc length.That is, there exists a unit-speed reparametrization of γ.Definition 2.33.A closed curve means a regular curve of the form : [, ] → ℝ  such that γ(a) = γ(b) and all derivatives match: () ′ = () ′ , () ′′ = () ′′ .If additionally is one-to-one on the domain [a, b), then it is called a simple closed curve ( Proposition 2.34.A regular curve : [, ] → ℝ  is a closed curve if and only if there exists a periodic regular curve :  → ℝ  with period ba such that  ̂ for all t ∈ [a, b].Definition 2.35.Let : [, ] → ℝ  be a closed curve.A reparametrization of γ is a function of the form  ̂ =   o φ: [c, d] → ℝ  , where  ∈ ℝ and φ: [c, d] → [a +, b +] is a smooth bijection with now here vanishing derivative, whose derivatives all match at c and d; that is, Proposition 2.38.If γ is parameterized by arc length, then κ = |a(t)|.Definition 2.39.Let :  → ℝ  be a regular curve.Define the unit tangent and the unit normal vectors at t ∈ I as () = () |()| , () =  ⊥ () | ⊥ ()| .Notation 2.43.The n-th degree of Taylor Polynomials of γ(x) is define as If γ : I → ℝ 3 is a regular space curve, then for all t ∈ I, Individually they are called the unit tangent, unit normal, and unit binormal vectors at t.