Chapter 6: Hyperbolic Analytic Geometry
... on each of these two perpendicular lines. By this we need to choose a positive and a negative direction on each line and a unit segment for each. There are other coordinate systems that can be used, but this is standard. We will call these the u-axis and the v-axis. For any point P ∈ H 2 let U and V ...
... on each of these two perpendicular lines. By this we need to choose a positive and a negative direction on each line and a unit segment for each. There are other coordinate systems that can be used, but this is standard. We will call these the u-axis and the v-axis. For any point P ∈ H 2 let U and V ...
Sample Geometry Projects with Scratch
... making a connection between the 360 degrees in a circle and the total number of degrees to turn in drawing each figure 2. Modify the script to use a variable for the side length and use it to also set the angle measurements. 3. Create a graphic art project which makes use of these concepts. ...
... making a connection between the 360 degrees in a circle and the total number of degrees to turn in drawing each figure 2. Modify the script to use a variable for the side length and use it to also set the angle measurements. 3. Create a graphic art project which makes use of these concepts. ...
Describing three-dimensional structures with spherical and
... approach for determining apparent dip. The first step is to determine the poles to our two planes. The second step is to express the pole vectors in Cartesian coordinates. (Alternatively — and this is how we will proceed — we can use the formulae above that give the N, E, D components directly in te ...
... approach for determining apparent dip. The first step is to determine the poles to our two planes. The second step is to express the pole vectors in Cartesian coordinates. (Alternatively — and this is how we will proceed — we can use the formulae above that give the N, E, D components directly in te ...
2.5 CARTESIAN VECTORS
... Using trigonometry, “direction cosines” are found using the formulas These angles are not independent. They must satisfy the following equation. cos ² α + cos ² β + cos ² γ = 1 This result can be derived from the definition of a coordinate direction angles and the unit vector. Recall, the formula fo ...
... Using trigonometry, “direction cosines” are found using the formulas These angles are not independent. They must satisfy the following equation. cos ² α + cos ² β + cos ² γ = 1 This result can be derived from the definition of a coordinate direction angles and the unit vector. Recall, the formula fo ...
Unit 4: Polygon Objectives
... 4. Solve multistep problems involving angle measure and side length in parallelograms, with an emphasis on problems in which angles measure or side length is written as a variable expression. (various sections) 5. Use coordinates to prove simple geometric theorems algebraically. For example, write c ...
... 4. Solve multistep problems involving angle measure and side length in parallelograms, with an emphasis on problems in which angles measure or side length is written as a variable expression. (various sections) 5. Use coordinates to prove simple geometric theorems algebraically. For example, write c ...
Honors Geometry Course Outline 2017
... cases a particular algorithm works. Scale: scale factors; invariant relationships between scaled copies; area and volume relationships between scaled copies; computing scale factors and using scale factors to compute lengths, areas, and volumes. (3a) Find areas and perimeters by dissection, developi ...
... cases a particular algorithm works. Scale: scale factors; invariant relationships between scaled copies; area and volume relationships between scaled copies; computing scale factors and using scale factors to compute lengths, areas, and volumes. (3a) Find areas and perimeters by dissection, developi ...
VECTOR ADDITION
... In this case we will start with two vectors at a time and draw them from the origin of our coordinate system drawn at the center of the graph paper. These two vectors will represent the adjacent sides of a parallel-ogram which is a figure whose opposite sides are parallel and equal. Then we will com ...
... In this case we will start with two vectors at a time and draw them from the origin of our coordinate system drawn at the center of the graph paper. These two vectors will represent the adjacent sides of a parallel-ogram which is a figure whose opposite sides are parallel and equal. Then we will com ...
4-8_Triangles_and_Coordinate_Proof
... Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base. The first step is to position and label an isosceles triangle on the coordinate plane. Place the base of the isosceles triangle along the ...
... Write a coordinate proof to prove that the segment that joins the vertex angle of an isosceles triangle to the midpoint of its base is perpendicular to the base. The first step is to position and label an isosceles triangle on the coordinate plane. Place the base of the isosceles triangle along the ...
Sample: Geometry - Scholl Geometry Exam 1
... If it is snowing in Dallas, Texas, then it is snowing in the United States. Answer: The statement is false. A counterexample may be “It is snowing in Dallas, Texas, but it is not snowing in Miami, Florida”. ...
... If it is snowing in Dallas, Texas, then it is snowing in the United States. Answer: The statement is false. A counterexample may be “It is snowing in Dallas, Texas, but it is not snowing in Miami, Florida”. ...
19 Orthogonal projections and orthogonal matrices
... components, the cosine of the angle between x and y is computed with the same formula as in the standard basis, and so on. We can summarize this by saying that Euclidean geometry is invariant under orthogonal transformations. Exercise: ** Here’s another way to get at the same result. Suppose A is a ...
... components, the cosine of the angle between x and y is computed with the same formula as in the standard basis, and so on. We can summarize this by saying that Euclidean geometry is invariant under orthogonal transformations. Exercise: ** Here’s another way to get at the same result. Suppose A is a ...
Answers
... We expand kv + wk2 using properties of the dot product: kv + wk2 = (v + w) · (v + w) =v·v+v·w+w·v+w·w = kvk2 + 2(v · w) + kwk2 (since v · w = w · v) = kvk2 + kwk2 (since v ⊥ w ⇔ v · w = 0) Thus, kv + wk2 = kvk2 + kwk2 , as promised. 12. Let A and B be endpoints of a diameter of a circle with a radi ...
... We expand kv + wk2 using properties of the dot product: kv + wk2 = (v + w) · (v + w) =v·v+v·w+w·v+w·w = kvk2 + 2(v · w) + kwk2 (since v · w = w · v) = kvk2 + kwk2 (since v ⊥ w ⇔ v · w = 0) Thus, kv + wk2 = kvk2 + kwk2 , as promised. 12. Let A and B be endpoints of a diameter of a circle with a radi ...
Curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are Cartesian, cylindrical and spherical polar coordinates. A Cartesian coordinate surface in this space is a plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R3 (for example, motion of particles under the influence of central forces) is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. One would for instance describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere. Spherical coordinates are one of the most used curvilinear coordinate systems in such fields as Earth sciences, cartography, and physics (in particular quantum mechanics, relativity), and engineering.