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College Mathematics Review for Test 3: Geometries

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College Mathematics Review for Test 3: Geometries College Mathematics review for Test 3: geometries Know the difference between length, area, and volume. Know the dimensionality of each measure. Know what units are used for each measure. Know how to calculate the area inside a square, rectangle, triangle, circle given appropriate information. Know the relationship between circumference and diameter of a circle. Know what it means for two objects to be (geometrically) similar. Know what scale factor (SF) means. Calculate the scale factor in a given situation involving similar triangles or other objects and use it to solve for an unknown length. Know how area, volume, and weight grow when shapes are enlarged or reduced without distortion. Given lengths of corresponding parts in two similar objects and the area/volume/weight of one of the objects, calculate the area/volume/weight of the other object. Know the definition of geodesic. Explain, in words and/or formulas, how Euclidean distance is measured. Given two points, calculate the Euclidean distance between them. Describe, in words, what geodesics in Euclidean geometry look like. Given two points, draw a Euclidean geodesic from one point to the other. Given two points, tell whether there is one or more than one geodesic joining them. Draw a Euclidean circle. Explain, in words and/or formulas, how taxicab distance is measured. Given two points, calculate the taxicab distance between them. Describe, in words, what geodesics in taxicab geometry look like. Given two points, draw a taxicab geodesic from one point to the other. Given two points, tell whether there is one or more than one geodesic joining them. Draw a taxicab circle. Explain, in words and/or formulas, how spherical distance is measured. Given two points, calculate the spherical distance between them. Describe, in words, what geodesics in spherical geometry look like. Given two points, draw a spherical geodesic from one point to the other. Given two points, tell whether there is one or more than one geodesic joining them. Draw a spherical circle. Describe situations in which the taxicab distance between two points is (more than, less than, equal to) the Euclidean distance between the points. Given two points, draw the set of points taxicab-between those points. Given two points, draw the taxicab bisector of those points. Given a point on a curve, draw the osculating circle at that point. Estimate the turning radius (TR) at a point. State the relationship between turning radius and curvature. Given two points, tell which one has higher curvature. Know the restrictions on the sum of angles in a Euclidean triangle and in a spherical triangle. Give a precise definition of each of the following terms: circle, circumference, diameter, radius, sphere, taxicab distance, taxicab circle, taxicab-between, taxicab bisector, Give five different precise descriptions of a great circle not involving the concept of curvature. Know how each of these surfaces is defined: plane, sphere, cylinder, Möbius strip, torus, Klein bottle, projective plane. Compare and contrast any pair of these surfaces. Draw a flat manifold ("square") diagram for each of the surfaces. Know how to play "5-in-a-row" on a plane, a flat torus, a flat Klein bottle. Calculate the connected sum of two or more surfaces. Calculate the product of two or more curves and/or surfaces. Know the definition of N-cube, including 0-cube, 1-cube, 2-cube, 3-cube, and 4-cube (hypercube). Calculate the number of vertices, edges, faces, cubes, hypercubes, etc. in any N-cube. Discuss the ideas of topology and dimension by using analogies such as Flatland. For example, explain what a Flatlander would see if a sphere were to pass through his/her plane. Then, describe what a 3-dimensional creature would see if a hypersphere were to pass through his/her 3-dim space. Understand that properties of these surfaces we discussed can be understood intrinsically, that is, from the point of view of an ant or "Flatlander" living on the surface. For example, we can imagine living on a torus and understanding it without curvature in some 3-dimensional space. By analogy, we can understand living in a 3-dimensional "torus" without the concept of curvature in a higher-dimensional space..
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