Polyhedral combinatorics has been a topic of interest in modern day’s computational geometry. The founding of Steinitz’s Theorem in 1922 revealed consequential relations between graph theory and polyhedral combinatorics. It allows us to better investigate on the topology of convex polyhedrons. In this paper, we proposed an algorithm that generates a unique sequence of points, using the vertices of a triangulated polyhedron, pre-determined by the selection of the starting 3 vertices in the sequence. Following that, we discover an interesting relation between the sequence and the volume of the polyhedron itself, in which we presented in the form of a sufficient condition. To further investigate which polyhedrons generate sequences that satisfy the sufficient condition, we study the problem in the context of graph theory, that is, the explorer walk (corresponding to the sequence of vertices) in maximal planar graphs (skeletons of triangulated convex polyhedrons). With that, we uncovered a family of maximal planar graphs, called the explorer graphs, which exhibits volumetric properties in the polyhedrons constructed from them, in regard to the explorer walk. In this paper, we also introduce generalized methods of constructing explorer graphs of higher order from explorer graphs of lower order, demonstrating the prevalence of explorer graphs. As the edges of a maximal planar graph is of great importance in tracing an explorer walk, we investigate on the line graph of maximal planar graphs, and re-establish a better definition of explorer graphs. Lastly, our paper covers the edge contraction of explorer graphs, which allows us to solve the volume of polyhedrons constructed from non-explorer graphs. For this, we presented a possible bound for the minimum number of edge contractions a non-explorer graph requires from an explorer graph. This will generalize the proposed method of finding volumes to any triangulated convex polyhedron.

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Cartographical projection is mathematical cartography. Map projection is the mathematical model of geoid. Gauss-Kruger projection is unable to realize seamless splicing between adjoining sheet maps. While Mercator projection, especially web Mercator projection, is able to realize seamless splicing but has larger measurement error. How to combine with the advantage of the two projections, apply the method of map projection transformation and realize accurate measurement under the framework of a global map, is the purpose of thisarticle. In order to demonstrate that the problem of measurement could be solved by map projection transformation, we studied the mathematical principle of Gauss - Kruger Projection and the Mercator Projection and designed the algorithm model based on characteristics of both projections. And established simulated data in ArcMap, calculated the error of the three indicators-the area, the length and the angle- under areas in different latitude(0°-4°、30°-34°、60°-64°)and longitude (120°-126°)range which validated the algorithm model. The conclusion suggests that it is feasible that using web Mercator Projection is capable to achieve a global map expression. Meanwhile, calculations of area, distance and angle using map projection transformation principle are feasible as well in Gauss - Kruger Projection. This algorithm combines the advantage of the two different kinds of projection, is able to satisfy uses' high demand and covers the shortage of measurement function of online map services in China. It has certain practical value.

In this paper we first introduce a fractional form formula among a number of Euler’s formulas. We then extend the formula and with mathematical induction prove the case when the number of terms increases and the exponent is integer. Afterwards, we study the connection between Euler’s formula and Lagrange interpolating polynomial and use the latter to prove part of the extended formula. We then obtain a new formula from this connection. At last, we derive a set of new equations from the extended formula.