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Geometry is one of the oldest branches of mathematics that is concerned with the shape, size, angles, and dimensions of objects in our day-to-day life. Geometry in mathematics plays a crucial role in understanding the physical world around us and has a wide range of applications in various fields, from architecture and engineering to art and physics.
There are two types of shapes in Euclidean Geometry: Two dimensional and Three-dimensional shapes. Flat shapes are 2D shapes in plane geometry that include triangles, squares, rectangles, and circles. 3D shapes in solid geometry such as a cube, cuboids, cones, and so on are also known as solids.
Fundamental geometry is based on points, lines, and planes, as described in coordinate geometry.
In this article, you will learn everything related to Geometry like what is geometry, branches of geometry, what are the different types of geometry, examples of geometry, etc.
<img alt="Geometry-Banner">
Table of Content
- What is Geometry?
- Geometry Definition in Maths
- Branches of Geometry
- Plane Geometry
- Angles in Geometry
- Polygon and its Types
- Circle in Geometry
- Solid Geometry
- Geometry Formulas
- Solved Examples on Geometry
What is Geometry?Geometry is the study of different varieties of shapes, figures, and sizes. It gives us knowledge about distances, angles, patterns, areas, and volumes of shapes.
The principles of geometry depend on points, lines, angles, and...
- Algebraic Geometry
- Discrete Geometry
- Differential Geometry
- Euclidean Geometry
- Non-Euclidean Geometry(Elliptical Geometry and Hyperbolic Geometry)
- Convex Geometry
- Topology
- A straight line can be drawn from one given point to another.
- The length of a straight line is infinite in both directions.
- Any specified point can serve as the circle’s center and any length can serve as the radius.
- All right angles are congruent.
- Any two straight lines that are equal in distance from one another at two points are infinitely parallel.
- The things that are equal to the same things are equal. If A = C and B = C then A = C
- If equals are added to equals, the wholes are equal. If A = B and C = D, then A + C = B + D
- If equals are subtracted, the remainders are equal.
- The coinciding things are equal t
- The whole is greater than its part. If A > B, then there exists C such that A = B + C.
- The things that are double the same are equal.
- The things that are halves of the same thing are equal
- Points – A point is the no-dimensional fundamental unit of geometry.
- Lines – A line is a straight path on a plane that extends in both directions with no endpoints.
- Angles – Plane geometry consists of lines, circles, and triangles of two dimensions. Plane geometry is another name for two-dimensional geometry.
- Collinear points are the ones that lie on the same line.
- A line segment is part of a line that has two endpoints and is finite in length.
- A ray is a line segment that extends indefinitely in one direction. A line has no endpoints.
- Line, line segment, and ray are different from each other.
- Acute Angle- An Angle between 0 to 90 degrees.
- Obtuse Angle– An angle more than 90 degrees but less than 180 degrees.
- Right Angle– An angle of 90 degrees.
- Straight Angle– An angle of 180 degrees is a straight line.
- Lines and Angles
- Pairs of Angles
- Triangles
- Quadrilaterals
- Pentagon
- Hexagon
- Heptagon
- Octagon
- Nonagon
- Decagon
- Polygons and their types
- Measures of the Exterior Angles of a Polygon
- Rectangle, Square, Rhombus, Parallelogram
- Some Special Parallelograms
- Parallel lines and a transversal
- Lines parallel to the same line and Angle Sum Property
- Properties of triangles
- Angle Sum Property of a Triangle
- Inequalities in a triangle
- Theorem – Angle opposite to equal sides of an isosceles triangle are equal
- Angle sum property of a quadrilateral
- Types of quadrilateral
- Properties of Parallelograms
- MidPoint Theorem
- Rhombus
- Kite – Quadrilaterals
- Area of 2D Shapes
- Figures on the same base and between the same parallels
- Circles and its Related Terms
- Circle Theorems
- Theorem – There is one and only one circle passing through three given non-collinear points
- Theorem – The sum of opposite angles of a cyclic quadrilateral is 180°
- Inscribed Shapes in a Circle
- Basic Construction
- Construction of Triangles
- Construction of a Quadrilateral
- Euclid’s Definitions, Axioms, and Postulates
- An equivalent version of Euclid’s Fifth postulate
- Euclid’s Division Algorithm
- Properties of Triangles
- Construction of Similar Triangles
- Similar Triangles
- Pythagoras Theorem and its Converse
- Thales’s Theorem
- Criteria for Similarity of Triangles
- Congruence of Triangles
- Faces
- Edges
- Vertices
- Visualizing Solid Shapes
- Mapping Space Around Us
- Cartesian Coordinate System
- Cartesian Plane
- Coordinate Geometry
- Distance formula
- Section formula
- Mid-point Formula
- Area of a Triangle
- Tangent to a circle
- A tangent at any point of a circle is perpendicular to the radius through the point of contact
- Number of Tangents from a point on a circle
- Lengths of tangents drawn from an external point to a circle are equal
- Division of Line Segment in Given Ratio
- Construction of tangents to a circle
- The perimeter of circular figures, Areas of sector and segment of a circle & Areas of a combination of plane figures
- Coordinate Axes and Coordinate Planes in 3D
- Distance Formula & Section Formula
- Slope of a Straight Line
- Introduction to Two-Variable Linear Equations in Straight Lines
- Forms of Two-Variable Linear Equations of a Line
- Point-slope Form
- Slope-Intercept Form of Straight Lines
- Writing Slope-Intercept Equations
- Standard Form of a Straight Line
- x-intercepts and y-intercepts of a Line
- Graphing slope-intercept equations
- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in 3D
- Angle between two lines
- Shortest Distance Between Two Lines in 3D Space
- Points, Lines, and Planes
- Linear Programming
- Graphical Solution of Linear Programming Problems
- Rectangle: Area = length × width
- Square: Area = side × side (or side²)
- Triangle: Area = ½ × base × height
- Circle: Area = π × radius²
- Rectangle: Perimeter = 2 × (length + width)
- Square: Perimeter = 4 × side
- Triangle: Perimeter = side₁ + side₂ + side₃
- Circle: Circumference = 2 × π × radius
- Cube: Volume = side × side × side (or side³)
- Rectangular Prism: Volume = length × width × height
- Cylinder: Volume = π × radius² × height
- Sphere: Volume = ⁴⁄₃ × π × radius³
- Sine (sin): sin(θ) = opposite / hypotenuse
- Cosine (cos): cos(θ) = adjacent / hypotenuse
- Tangent (tan): tan(θ) = opposite / adjacent
- Triangles in Geometry
- Geometry and Co-ordinates
- Algebraic Geometry
- Discrete Geometry
- Differential Geometry
- Euclidean Geometry
- Non Euclidean Geometry(Elliptical Geometry and Hyperbolic Geometry)
- Convex Geometry
- Topology
- Euclidean Geometry: Explores plane and solid figures through axioms and theorems.
- Differential Geometry: Extends calculus principles, crucial in physics for understanding curves and spaces.
- Algebraic Geometry: Focuses on curves and surfaces, utilizing linear and polynomial algebraic equations.
- Discrete Geometry: Analyzes relative positions of basic geometric objects.
- Analytic Geometry: Studies geometric figures and constructions using coordinate systems.
- Riemannian Geometry: Encompasses non-Euclidean geometries, offering diverse geometric perspectives.
- Complex Geometry: Investigates geometric structures based on the complex plane.
- Computational Geometry: Examines properties of explicitly defined algebraic varieties, vital in computational mathematics and computer science.
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Geometry is one of the oldest branches of mathematics that is concerned with the shape, size, angles, and dimensions of objects in our day-to-day life. Geometry in mathematics plays a crucial role in understanding the physical world around us and has a wide range of applications in various fields, from architecture and engineering to art and physics.
There are two types of shapes in Euclidean Geometry: Two dimensional and Three-dimensional shapes. Flat shapes are 2D shapes in plane geometry that include triangles, squares, rectangles, and circles. 3D shapes in solid geometry such as a cube, cuboids, cones, and so on are also known as solids.
Fundamental geometry is based on points, lines, and planes, as described in coordinate geometry.
In this article, you will learn everything related to Geometry like what is geometry, branches of geometry, what are the different types of geometry, examples of geometry, etc.
<img alt="Geometry-Banner">
Table of Content
Polygon and its...
- What is Geometry?
- Geometry Definition in Maths
- Branches of Geometry
- Plane Geometry
- Angles in Geometry
- Polygon and its Types
2/5
Geometry is one of the oldest branches of mathematics that is concerned with the shape, size, angles, and dimensions of objects in our day-to-day life. Geometry in mathematics plays a crucial role in understanding the physical world around us and has a wide range of applications in various fields, from architecture and engineering to art and physics.
There are two types of shapes in Euclidean Geometry: Two dimensional and Three-dimensional shapes. Flat shapes are 2D shapes in plane geometry that include triangles, squares, rectangles, and circles. 3D shapes in solid geometry such as a cube, cuboids, cones, and so on are also known as solids.
Fundamental geometry is based on points, lines, and planes, as described in coordinate geometry.
In this article, you will learn everything related to Geometry like what is geometry, branches of geometry, what are the different types of geometry, examples of geometry, etc.
<img alt="Geometry-Banner">
Table of Content
Polygon and its...
- What is Geometry?
- Geometry Definition in Maths
- Branches of Geometry
- Plane Geometry
- Angles in Geometry
- Polygon and its Types
3/5
Top Answer
Answered Apr 18, 2019 · 9 votes
math_sin is defined via the FUNC1
macro:
FUNC1(sin, sin, 0, "sin($module, x, /)\n--\n\n" "Return the sine of x (measured in radians).")-
where FUNC1
is defined as:
#define FUNC1(funcname, func, can_overflow, docstring) \ static PyObject * math_##funcname(PyObject *self, PyObject *args) { \ return math_1(args, func, can_overflow); \ }\ PyDoc_STRVAR(math_##funcname##_doc, docstring);
so the preprocessor expands this to:
static PyObject * math_sin(PyObject *self, PyObject *args) { return math_1(args, sin, 0); } PyDoc_STRVAR(math_sin_doc, "sin($module, x, /)\n--\n\n" "Return the sine of x (measured in radians).");
(but then all on one line, and with the PyDoc_STRVAR macro also having been expanded)
So math_sin(module, args) basically is a call to math_1(args, sin, 0), and math_1(args, sin, 0)
calls math_1_to_whatever(args, sin, PyFloat_FromDouble, 0)
which takes care of validating that a Python float was passed in, converting that to a C double, calling sin(arg_as_double), raising exceptions as needed or wrapping the double return value from sin() with the PyFloat_FromDouble function passed in by math_1() before returning that result to the caller.
sin() here is the double sin(double x)
function defined in POSIX math.h
.
You can, in principle, preprocess the whole Python source tree and dump the output into a new directory; the following does presume you successfully built the python binary already, as it is used to extract the necessary include flags for gcc:
find . -type d -exec mkdir -p /tmp/processed/{} \;(export FLAGS=$(./python.exe -m sysconfig | grep PY_CORE_CFLAGS | cut -d\" -f2) && \ find . -type f \( -name '*.c' -o -name '*.h' \) -exec gcc -E $FLAGS {} -o /tmp/processed/{} \;)-
and then math_sin will show up in /tmp/preprocessed/Modules/mathmodule.c.
Or you can tell the compiler to save preprocessor output to .i files with the -save-temps flag:
make clean && make CC="gcc -save-temps"-
and you'll find make_sin in Modules/mathmodule.i.
4/5
Top Answer
Answered Mar 01, 2018 · 3 votes
It is now 2.5 years after the OP asked this question, so my answer is probably more for anyone who has followed a link here, hoping for some insight. On the grounds that FEM programming is special,0 I will try to answer this question rather than flag it as off-topic. Anyway, some of my answer is applicable to FEM in general, some is specific to Abaqus.
Quick check: If you're only asking for the specific numerical value to use for the (usual or standard) location of integration points, then the answer is that it depends. Luckily, standard values are widely available for a variety of elements (see resources below).
However, I assume you're asking about writing a User-Element (UEL) subroutine but are not yet familiar with how elements are formulated, or what an integration point is.
The answer: In the standard displacement-based FEM the constitutive response of an individual finite element is usually obtained by numerical integration (aka quadrature) at one or more points on or within the element. How many and where these points are located depends on the element type, certain performance tradeoffs, etc, and the particular integration technique being used. Integration techniques that I have seen used for continuum (solid) finite elements include:
- More Common: Gauss integration -- the number & position of sampling points are determined by the Gauss quadrature rule used; nodes are not included in the sampling domain of [-1,1].
- Less Common: Newton-Cotes integration -- evenly spaced sampling points; includes the nodes in the sampling domain of (-1,1).
In my experience, the standard practice by far is to use Gauss quadrature or reduced integration methods (which are often variations of Gauss quadrature). In Gauss quadrature, the location of the integration points are taken at special ("optimal") points within the element known as Gauss points which have been shown to provide a high level of reliably accurate solutions for a given level of computational expense - at least for the typical polynomial functions used for many isoparametric finite elements. Other integration techniques have been found to be competitive in some cases1 but Gauss quadrature is certainly the gold standard. There are other techniques that I'm not familiar with.
Practical advice: Assuming an isoparametric formulation, in the UEL you use "element shape functions" and the primary field variables defined by the nodal degrees of freedom (with a solid mechanics focus, these are typically the displacements) to calculate the element strains, stresses, etc. at each integration point. If this doesn't make sense to you, see resources below.
Note that if you need the stresses at the nodes (or at any other point) you must extrapolate them from the integration points, again using the shape functions, or calculate/integrate directly at the nodes.
Suggested resources: Please: If you're writing a user subroutine you should already know what an integration point is. I'm sorry, but that's just how it is. You have to know at least the basics before you attempt to write a UEL.
That said, I think it's great that you're interested in programming for FEA/FEM. If you're motivated but not at university where you can enroll in an FEM course or two, then there are a number of resources available, from Massive Open Online Courses (MOOCs), to a plethora of textbooks - I generally recommend anything written by Zienkiewicz. For a readable yet "solid" introduction with an emphasis on solid mechanics, I like Concepts and Applications of Finite Element Analysis, 4th Edition, by Cook et al (aka the "Cook Book"). Good luck!
0 You typically need a lot of background before you even ask the right questions.
1 Trefethen, 2008, "Is Gauss Quadrature Better than Clenshaw-Curtis?", DOI 10.1137/060659831
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mathandreadinghelp.org › 4th_grade_geometry_vocabulary4th Grade Geometry Vocabulary: Essential Words and Definitions
mathandreadinghelp.org › 4th_grade_geometry_vocabulary2 days ago · Essential 4th Grade Geometry Vocabulary. A point indicates an exact location. Often, a point is seen as a dot on a graph. A line is a 2-D object that is straight and infinitely long. Horizontal lines go left and right, and vertical lines travel up and down. A ray is a part of a line that begins at a defined point and continues infinitely.
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thirdspacelearning.com › blog › maths-masteryMaths Mastery Toolkit: A Practical Guide To Mastery Teaching ...
thirdspacelearning.com › blog › maths-masteryMay 23, 2024 · Use this maths mastery toolkit of guidance and mastery resources to support staff and teachers in your school to deepen their understanding of teaching for mastery in maths. Written by Third Space Learning’s own maths mastery expert Wendy Liu this maths mastery toolkit of resources and techniques to use in your lessons emphasises the ...
www.geeksforgeeks.org › cartesian-planeCartesian Plane: Definition, Axes, Origin, Quadrants & Examples
www.geeksforgeeks.org › cartesian-planeMay 16, 2024 · Origin. Cartesian Plane Quadrants. Coordinate of a Point. Plotting Points on Cartesian Plane. One Dimensional Plane (Line) Three Dimensional Plane. Cartesian Plane Definition. A two-dimensional coordinate plane system that is formed by intersecting two perpendicular lines is called the cartesian plane.
flexbooks.ck12.org › cbook › ck-12-precalculusCK12-Foundation
flexbooks.ck12.org › cbook › ck-12-precalculusMay 16, 2024 · When you intersect a plane with a two sided cone so that the plane passes vertically through the central point of the two cones, it produces a degenerate hyperbola. Example 2. Transform the conic equation into standard form and sketch. 0 x 2 + 0 x y + 0 y 2 + 2 x + 4 y − 6 = 0. This is the line y = − 1 2 x + 3 2. Example 3
mathandreadinghelp.org › 8th_grade_math_vocabulary8th Grade Math Vocabulary | Terms and Definitions
mathandreadinghelp.org › 8th_grade_math_vocabulary3 days ago · A shorthand method for writing very small or very large numbers using positive and negative powers of ten. For example, 7,000,000 can be written as 7.0 x 10^6, and 0.00007 is written as 7.0 x 10^-5. Slope. The steepness of a line on a graph. It is calculated by the change in y divided by the change in x between two coordinate points ( x, y) on ...
libraryguides.centennialcollege.ca › cMath help from the Learning Centre - Centennial College
libraryguides.centennialcollege.ca › c6 days ago · Graphing Definitions. The graph of an equation in two variables x and y is the set of points in the xy-plane whose coordinates are members of the solution set of that equation. In other words, a graph of an equation is made up of all the points (i.e. x and y values) that satisfy that equation. In the Cartesian system, each point on the graph is ...
www.mathmammoth.com › early_geometryMath Mammoth Early Geometry: geometry concepts for grades 1-3
www.mathmammoth.com › early_geometryMay 27, 2024 · Math workbook for first, second or third grade, Math Mammoth Early Geometry deals with common shapes, breaking them apart or forming new ones, the concept of right angle, symmetry, and an introduction to area, perimeter, and solids.
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