Miller started with the Mercator projection and moved the latitude lines closer to the equator. The distance L in the Mercator projection between each parallel and the equator was measured, and the parallel was moved to a distance of 0 . 8 L from the equator. Thus, near the equator, this projection is virtually identical to Mercator. Another result of this shrinking of distances is that the height of the lines of longitude (the meridians) is 0.73 the length of the latitudes. Each pole, which on the Earth is a point, is displayed in this projection as a line of latitude, thereby causing maximum distortions at the poles. The mathematical expression of this projection starts with a point with longitude and latitude on the sphere. The point is mapped by this projection to the point x = , y = 5 4 ln tan 4 + 2 5 = 5 4 sinh 1 tan 4 5 on the **map** (where is the longitude at the center of the **map**). 220 4 Nonlinear Projections The Miller cylindrical projection is often selected by cartographers for atlas **maps** of the **world** instead of the more popular Mercator projection. Evidently, some mapping experts feel that this variant is somewhat more appropriate or is simply more pleasing to the eye. Nonlinear: Behaving in an erratic and unpredictable fashion, unstable. When used to describe the behavior of a machine or program, it suggests that said machine or program is being forced to run far outside of design specications. Eric Raymond, The Jargon File (1997) A Vector Products It is trivial to add and subtract vectors, but vectors can also be multiplied. This short appendix is a reminder of (or a refresher on) the two important operations of dot product and cross product. The dot product (or inner product) of two vectors is denoted by P Q and is dened as the scalar ( P x ,P y ,P z )( Q x ,Q y ,Q z ) T = PQ T = P x Q x + P y Q y + P z Q z . This simple denition implies that the dot product is commutative, P Q = Q P , and is also distributive with respect to vector addition or subtraction, P ( Q T ) = P Q P T . The dot product also has a simple and useful geometric interpretation, it equals | P || Q | cos , where is the angle between the vectors. The dot product of perpendicular (or orthogonal ) vectors is therefore zero. We use Figure A.1 to prove this interpretation. Part a shows a triangle with three sides a , b , and c and three angles A , B , and C opposite those sides. We draw a line from vertex B that is perpendicular to side b . This line divides the triangle into two right-angle triangles. The three sides of the triangle on the right are a , a sin C , and a cos C , while the sides of the triangle on the left are c , a sin C , and b a cos C . Applying Pythagoras’s theorem to the latter triangle yields the law of cosines c 2 = ( a sin C ) 2 + ( b a cos C ) 2 = a 2 sin 2 C + b 2 2 ab cos C + a 2 cos 2 C = a 2 (sin 2 C + cos 2 C ) + b 2 2 ab cos C = a 2 + b 2 2 ab cos C. C…. Type your question here! Please include all relevant details, attachments, and requirements so your tutor can provide a complete answer. Source.