Complex numbers via rigid motions
... Now I is like multiplying the coodinates by 1 and H is like multiplying the coordinates by -1 This is not too outrageous as a dilation can be seen as a multiplication of the coordinates by a number <> 1 So, continuing into uncharted territory, we have H squared = 1 (fits with (-1)*(-1) = 1 and Q squ ...
... Now I is like multiplying the coodinates by 1 and H is like multiplying the coordinates by -1 This is not too outrageous as a dilation can be seen as a multiplication of the coordinates by a number <> 1 So, continuing into uncharted territory, we have H squared = 1 (fits with (-1)*(-1) = 1 and Q squ ...
Cylindrical and Quadric Surfaces
... Example: What is y = x? More correctly what is {(x, y, z) ∈ R3 : y = x}? It’s a plane. What about y = x2 ? Its a cylinder surface. What about y − z = x2 Again a cylinder surface but, in planes parallel to the xy–plane there is a parabola that moves up the y axis as you move up (positive z direction) ...
... Example: What is y = x? More correctly what is {(x, y, z) ∈ R3 : y = x}? It’s a plane. What about y = x2 ? Its a cylinder surface. What about y − z = x2 Again a cylinder surface but, in planes parallel to the xy–plane there is a parabola that moves up the y axis as you move up (positive z direction) ...
Direct visuomotor transformations for reaching (Buneo et al.)
... Transforming target location from eye- to head- to body-centered coordinates and then subtracting the body centered position of the hand ...
... Transforming target location from eye- to head- to body-centered coordinates and then subtracting the body centered position of the hand ...
Name: Period: Date: Unit 1: Introduction to Geometry Section 1.3
... Directions: Find the coordinates of the midpoint of a segment having the given endpoints. 1. E(-2, 6), F(-9, 3) ...
... Directions: Find the coordinates of the midpoint of a segment having the given endpoints. 1. E(-2, 6), F(-9, 3) ...
fpp revised
... We start with x1 , it has a non-zero coefficient in at least one y and one z, assume these are y1 and z1 . If it has different coefficients in these two expressions, we impose the condition: y1 = z1 , otherwise, we impose: y1 = −z1 . Here is an implicit example: z1 = a11 x1 + a21 x2 + . . . + aN 1 x ...
... We start with x1 , it has a non-zero coefficient in at least one y and one z, assume these are y1 and z1 . If it has different coefficients in these two expressions, we impose the condition: y1 = z1 , otherwise, we impose: y1 = −z1 . Here is an implicit example: z1 = a11 x1 + a21 x2 + . . . + aN 1 x ...
Algebra Notes
... Claim: The point (x, y) is a constructible point if and only if x and y are constructible numbers. Proof: ⇒: If (x, y) is a constructible point, draw the lines through it that are parallel to the x and y axes. These lines will intersect the axes at the points (x, 0) and (0, y), which are at distance ...
... Claim: The point (x, y) is a constructible point if and only if x and y are constructible numbers. Proof: ⇒: If (x, y) is a constructible point, draw the lines through it that are parallel to the x and y axes. These lines will intersect the axes at the points (x, 0) and (0, y), which are at distance ...
hyperbolic plane
... Hyperbolic Trigonometry and Euclidean Trigonometry: Differences and Similarities Trigonometry is the study between angles and sides of a triangle. The hyperbola comes from the conic section family. The conic section has four shapes; those are the circle, the ellipse, the parabola and the hyperbola. ...
... Hyperbolic Trigonometry and Euclidean Trigonometry: Differences and Similarities Trigonometry is the study between angles and sides of a triangle. The hyperbola comes from the conic section family. The conic section has four shapes; those are the circle, the ellipse, the parabola and the hyperbola. ...
3-5 Lines in the coordinate plane M11.B.2 2.3.11.A Objectives: 1)To
... Example: Transforming from Standard Form to Slope-Intercept Form ...
... Example: Transforming from Standard Form to Slope-Intercept Form ...
H10
... Remark. One can also show that every finitely generated flat module is projective, so for finitely generated A-modules flat and projective are equivalent. However, there are (non-finitely generated) flat modules which are not projective. For example, the ring Z[ 12 ] is a flat module over Z, but not ...
... Remark. One can also show that every finitely generated flat module is projective, so for finitely generated A-modules flat and projective are equivalent. However, there are (non-finitely generated) flat modules which are not projective. For example, the ring Z[ 12 ] is a flat module over Z, but not ...
Document
... Is any set of ordered pairs. The set of first coordinates in the ordered pairs is the domain of the relation, and ...
... Is any set of ordered pairs. The set of first coordinates in the ordered pairs is the domain of the relation, and ...
Homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.