Abstract: Systems and methods of automatically distilling concepts from math problems and dynamically constructing and testing the creation of math problems from a collection of math concepts comprising: providing a user interface to a user; receiving as input: a math problem; one or more math concepts; and/or a user data packet; extracting and compiling a concept cloud of one or more CLIs that comprise the mathematical concepts embodied in the input, describe the operation of the one or more math concepts, or relate to the UDP, respectively; generating one or more math problem building blocks from the concept cloud CLIs; applying a mathematical rules engine to the one or more math problem building blocks to build one or more additional math problems; and returning to the user, through the user interface, the one or more additional math problems built from the CLIs that define the concept cloud extracted from the input.

Abstract: Systems and methods of automatically distilling concepts from math problems and dynamically constructing and testing the creation of math problems from a collection of math concepts comprising: providing a user interface to a user; receiving as input: a math problem; one or more math concepts; and/or a user data packet; extracting and compiling a concept cloud of one or more CLIs that comprise the mathematical concepts embodied in the input, describe the operation of the one or more math concepts, or relate to the UDP, respectively; generating one or more math problem building blocks from the concept cloud CLIs; applying a mathematical rules engine to the one or more math problem building blocks to build one or more additional math problems; and returning to the user, through the user interface, the one or more additional math problems built from the CLIs that define the concept cloud extracted from the input.

Abstract: Systems and methods of automatically distilling concepts from math problems and dynamically constructing and testing the creation of math problems from a collection of math concepts comprising: providing a user interface to a user; receiving as input: a math problem; one or more math concepts; and/or a user data packet; extracting and compiling a concept cloud of one or more CLIs that comprise the mathematical concepts embodied in the input, describe the operation of the one or more math concepts, or relate to the UDP, respectively; generating one or more math problem building blocks from the concept cloud CLIs; applying a mathematical rules engine to the one or more math problem building blocks to build one or more additional math problems; and returning to the user, through the user interface, the one or more additional math problems built from the CLIs that define the concept cloud extracted from the input.

Abstract: Systems and methods of automatically distilling concepts from math problems and dynamically constructing and testing the creation of math problems from a collection of math concepts comprising: providing a user interface to a user; receiving as input: a math problem; one or more math concepts; and/or a user data packet; extracting and compiling a concept cloud of one or more CLIs that comprise the mathematical concepts embodied in the input, describe the operation of the one or more math concepts, or relate to the UDP, respectively; generating one or more math problem building blocks from the concept cloud CLIs; applying a mathematical rules engine to the one or more math problem building blocks to build one or more additional math problems; and returning to the user, through the user interface, the one or more additional math problems built from the CLIs that define the concept cloud extracted from the input.

Abstract: A method and an apparatus render on a display screen (14) an image of a three-dimensional object. The object is represented by image data arranged in a constructive solid geometry format including at least one halfspace that divides an object space (28) into an interior region that lies inside the halfspace and an exterior region that lies outside the halfspace. The boundary of the halfspace is defined by an implicit mathematical function. The method includes subdividing the object space into volume elements of either a cubical or rectangular parallelepiped shape. For each volume element, an upper bound and a lower bound are calculated to the values of each function defining the boundary of a halfspace. In a first preferred embodiment, the calculation of the upper and lower bounds of the function defining a halfspace within a volume element identifies all of the extrema of the function that lie in or on the boundary of the volume element.