Hyperbolic Tree

Nov 22, 2012

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Hyperbolic Tree

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Applications of Hyperbolic Functions

Applications of Hyperbolic Functions. Greg Kelly, Hanford High School, Richland, Washington. A hanging cable makes a shape called a catenary . Even though it looks like a parabola, it is not a parabola!. (for some constant a ). Length of curve calculation:.

1.44k views • 8 slides

Hyperbolic & Inverse Hyperbolic Functions

Hyperbolic & Inverse Hyperbolic Functions. Lesson 7.8. Catenary Curve. The curve formed by a hanging cable is called a catenary They behave similar to trig functions They are related to the hyperbola in similar manner as trig functions to the circle Thus are called hyperbolic functions.

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Hyperbolic Geometry

Hyperbolic Geometry. Chapter 9. Hyperbolic Lines and Segments. Poincaré disk model Line = circular arc, meets fundamental circle orthogonally Note: Lines closer to center of fundamental circle are closer to Euclidian lines Why?. Poincaré Disk Model. Model of geometric world

3.89k views • 41 slides

Hyperbolic Geometry Axiom

Hyperbolic Geometry Axiom. Hyperbolic Axiom (HA): In hyperbolic geometry there exist a line l and point P not on l such that at least two distinct lines parallel to l pass through P. Lemma 6.1: Rectangles do not exist. Hyperbolic Geometry UHT.

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Hyperbolic Geometry

Hyperbolic Geometry. Chapter 6. THE Quadrilateral!. Hyperbolic Geometry. Negation of Hilbert’s Euclidean Parallel Postulate: There exist a line l and point P not on l such that at least two distinct lines parallel to l pass through P. Basic Theorem 6.1: .

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Hyperbolic Functions

Hyperbolic Functions. The Hyperbolic Sine, Hyperbolic Cosine & Hyperbolic Tangent. The Inverse Hyperbolic Cotangent, Hyperbolic Secant & Hyperbolic Cosecant. Values. Remember Logarithmic Identities ( Calculus I ). Logarithmic Identities II. Identities. Proofs Identity (1).

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7.7 Hyperbolic Functions

7.7 Hyperbolic Functions. Math 6B Calculus II. Definition of Hyperbolic Functions. Hyperbolic Cosine (cosh) Hyperbolic Sine (sinh) Other Hyperbolic Functions. Hyperbolic Identities. sinh (- x ) = - sinh x cosh (- x ) = cosh x cosh 2 x – sinh 2 x = 1 1 – tanh 2 x = sech 2 x

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HYPERBOLIC APPROACH

Giuseppe “Bepi” Colombo. 1 st D V Rec (~1 m/s). 1 st D V Rec (~1 m/s). To the Sun. To the Sun. 2 nd D V Rec (~3 m/s). 2 nd D V Rec (~40 m/s). 3 rd D V Rec (~5 m/s). A DVANCED T OPICS IN A STRODYNAMICS Barcelona, July 5-10, 2004 USE OF GRAVITATIONAL CAPTURE

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5.9 Hyperbolic Functions

5.9 Hyperbolic Functions. Graph the following two functions:. These functions show up frequently enough that they have been given names. The behavior of these functions shows such remarkable parallels to trig functions, that they have been given similar names. Definitions.

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Hyperbolic Functions

Photo by Vickie Kelly, 2005. Greg Kelly, Hanford High School, Richland, Washington. Hyperbolic Functions. Scotty’s Castle, Death Valley, CA. Consider the following two functions:. These functions show up frequently enough that they have been given names.

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Hyperbolic Equations

Hyperbolic Equations. IVP:. u ( x ,0) = f ( x ), -  < x < . IBVP:. u ( x ,0) = f ( x ), 0 < x <  and u (0, t ) = g ( t ), t > 0. IVP. Method of Characteristics. Let C be a characteristic curve described by x = x ( t ). On C u ( x , t ) = u ( x ( t ), t ).

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Hyperbolic Heat Equation

1-D BTE. or. assume. (4.64). (4.63a). or. Hyperbolic Heat Equation. Integrate over velocity space assuming that t is an averaged relaxation time. Energy equation without convection. In 3-D. Cattaneo equation. Maxwell-Cattaneo equation. Decay in amplitude:. hyperbolic equation.

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Hyperbolic FUNCTIONS

Hyperbolic FUNCTIONS. HYPERBOLIC FUNCTIONS. Certain combinations of the exponential functions e x and e -x arise so frequently in mathematics and its applications that they are given special names. HYPERBOLIC FUNCTIONS.

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Hyperbolic Functions

Hyperbolic Functions. Dr. Farhana Shaheen Yanbu University College KSA. Hyperbolic Functions. Vincenzo Riccati (1707 - 1775) is given credit for introducing the hyperbolic functions. Hyperbolic functions are very useful in both mathematics and physics.

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Hyperbolic Equations

Hyperbolic Equations. Pure IVP:. Lax-Wendroff method. Preliminaries: Input: Integer p, lower spatial index at t jmax Integer q, upper spatial index at t jmax Integer jmax, number of time steps real k, time step real h, space step real a, coefficient

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5.10 Hyperbolic Functions

5.10 Hyperbolic Functions. Greg Kelly, Hanford High School, Richland, Washington. Objectives. Develop properties of hyperbolic functions. Differentiate and integrate hyperbolic functions. Develop properties of inverse hyperbolic functions.

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§23.1 Hyperbolic Models

§23.1 Hyperbolic Models. • A. Intro to Poincare's Disk Model. Point – Any interior point of circle C (the ordinary points of H or h-points). Line – Any diameter of C or any arc of a circle orthogonal to C in H are h-lines. 1. Poincare's Disk Model.

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Hyperbolic Geometry

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Hyperbolic Trees

Hyperbolic Trees. A Focus + Context Technique http://www.acm.org/sigchi/chi95/proceedings/papers/jl_bdy.htm John lamping Ramana Rao Peter Pirolli Joy Mukherjee. Essentials. Visualizing Hierarchies Features - More space to a part - Still maintain context Scheme

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Hyperbolic Functions

Hyperbolic trigonometric functions provide a useful notation for describing hyperbola parametrically. Idea. How do these definition relate to the complex exponential definitions of sin and cos ?. ?. Hyperbolic Functions. Def. Hyperbolic Sine. “shine”. Hyperbolic Cosine. “cosh”.

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Hyperbolic Trees

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Differentiating Hyperbolic Functions

The derivatives of hyperbolic functions follow from their exponential definitions. Idea. ?. Differentiating Hyperbolic Functions. Ex. Differentiating Hyperbolic Functions. Ex. Ex. Differentiating Hyperbolic Functions. Ex. Page 48 Exercise 3A.

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Presentation Transcript

Applications of Hyperbolic Functions

Hyperbolic &amp; Inverse Hyperbolic Functions

Hyperbolic Geometry

Hyperbolic Geometry Axiom

Hyperbolic Geometry

Hyperbolic Functions

7.7 Hyperbolic Functions

HYPERBOLIC APPROACH

5.9 Hyperbolic Functions

Hyperbolic Functions

Hyperbolic Equations

Hyperbolic Heat Equation

Hyperbolic FUNCTIONS

Hyperbolic Functions

Hyperbolic Equations

5.10 Hyperbolic Functions

§23.1 Hyperbolic Models

Hyperbolic Geometry

Hyperbolic Trees

Hyperbolic Functions

Hyperbolic Trees

Differentiating Hyperbolic Functions

Hyperbolic & Inverse Hyperbolic Functions