MAT119 - PowerPoint Presentation

Slide1

MAT119

Asst. Prof. Ferhat PAKDAMAR (Civil Engineer)N Blok 1-17 pakdamar@gtu.edu.tr

Gebze Technical UniversityDepartment of Architecture

Fall – 2016_2017

Week 2Slide2

Osmanlı Geometri- Üç dılı

birbirine müsavi müselleslerin irtifaını nasıl bulurlar.- Dılın murabbaından, dılın nısfının murabbaını nakşeder, kök murabbaını alırsın. -Kaim zaviyeli

müselleste, bir kaim zaviyenin karşısındaki kaim dılın kaim vetere nispetine o hadde zaviyenin nesi derler - Ceybi derlerSlide3

Recall

TrigonometrySlide4

Recall TrigonometrySlide5

RADYANSlide6

Necessity of Geometry

If you don’t want to

yaw from your route, you need geometry!

BACSlide7

History of Geometry

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", 

-metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales(6th Century BC).

By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia.Slide8

Basics of Geometry

Points, Lines &

PlanesSegments, Rays &

LinesDistance Between Points Distance Formula in “n” Dimensions

Angles Types of AnglesSlide9

Types,

methodologies and terminologies of GeometryAbsolute geometry

Affine geometryAlgebraic geometryAnalytic geometryArchimedes' use of infinitesimalsBirational geometryComplex geometryCombinatorial geometryComputational geometryConformal geometryConstructive

solid geometryContact geometryConvex geometryDescriptive geometryDifferential geometryDigital geometryDiscrete geometryDistance geometryElliptic geometryEnumerative geometryEpipolar geometryFinite geometryFractal geometryGeometry of numbersHyperbolic geometryIncidence geometryInformation geometryIntegral geometryInversive geometryInversive ring geometryKlein geometryLie sphere geometry

Non-Euclidean geometryNumerical geometryOrdered geometryParabolic geometryPlane geometryProjective geometryQuantum geometryReticular geometryRiemannian geometryRuppeiner geometrySpherical geometrySymplectic geometrySynthetic geometrySystolic geometryTaxicab geometryToric geometryTransformation geometryTropical geometry… Slide10

Fractal

Geometry1A geometric figure that appears irregular at all scales of length, e.g. a fern2

A geometric figure which has a Hausdorff dimension which is greater than its topological dimension3Having the form of a fractal4A mathematically generated pattern that is endlessly complex Fractal patterns often resemble natural phenomena in the way they repeat elements with slight variations each time

5A kind of image that is defined recursively, so that each part of the image is a smaller version of the whole6A fractal is a shape where self-similarity dimension is greater than topological dimension7A geometric entity characterized by self-similarity (see figure 2): the whole entity is similar to a smaller portion of itself, but has a higher level of recursion (see recursion) Therefore, it can usually be represented by a recursive definition When using a fractal to represent a physical object, some degree of randomness is usually added to make the image more realistic8groups that have broken dimensions so that each one looks like an exact copy of the second (like the Mandelbrot group in Mathematics); (In Computers) geometric shapes that have interesting contour lines9A geometric figure that repeats itself under several levels of magnification, a shape that appears irregular at all scales of length, e.g. a fern10A geometric figure, built up from a simple shape, by generating the same or similar changes on successively smaller scales; it shows self-similarity on all scalesSlide11

Fractal

Geometry

Every

fractal is a pattern but every pattern is not a fractalA pattern can be a fractal with these rules1- Pattern

must be scaled 2- Previous form must be contained3-

Must

proceed

according

to

a

specific

rule

It

is a

pattern

.

Because

Next

shape

can be

predicted

Not a

fractal

Because

Shape

is not scaled

Pattern

?

Fractal

?Slide12

Fractal

Geometry

Every

fractal is a pattern but every pattern is not a fractalA pattern can be a fractal with these rules1-

Pattern must be scaled 2- Previous form must be contained

3-

Must

proceed

according

to

a

specific

rule

It

is a

pattern

.

Because

Next

shape

can be

predicted

Not a

fractal

Because

Next shape is not

encapsulate

the

previous

Pattern

?

Fractal

?Slide13

Fractal

Geometry

Every

fractal is a pattern but every pattern is not a fractalA pattern can be a fractal with these rules1-

Pattern must be scaled 2- Previous form must be contained

3-

Must

proceed

according

to

a

specific

rule

It

is a

pattern

and

a

fractal

Pattern

?

Fractal

?Slide14

Fractal

GeometrySlide15

Fractal

GeometrySlide16

Fractal

GeometrySlide17

Fractal

GeometrySlide18

Dimension of a Fractal (Hausdorff)

D:

Dimension of a fractalN: Number of repetitions (total): Scaling factorWhat does D describe

? Slide19

Sample

Fractal FiguresThese figures are

very important for midterm exam and Homeworks !Slide20

Basics of Geometry

Points, Lines &

PlanesSegments, Rays &

LinesDistance Between Points Distance Formula in “n” Dimensions

Angles Types of Angles