![]() ![]() Generate ideas for possible solutions, individually and through group activity apply age-appropriate mathematics and science skills evaluate the ideas and determine the best solution and explain reasons for the choices Investigate prior solutions and ideas from books, magazines, family, friends, neighbors, and community members The four instances of the impossible torus differ from each other only by the amount the square cross section is rotated while swept along the 360 degrees of the circle, starting from 90 degrees rotation on the left, all the way to 360 degree in the right example.ĭescribe objects, imaginary or real, that might be modeled or made differently and suggest ways in which the objects can be changed, fixed, or improved The Impossible Torus demonstrates sweeping a square shape along a circle trajectory, while the square is rotated. Rotation along with sweep can be combined to give a twist to the generated surface Under this definition, curves C1(u) and C2(v) are referred to as the profile curve and trajectory curve.Ī square or rectangle swept along a straight pathĪ triangle swept along straight path yields wedgeĪ circle along a straight path results in cylinderĪ circle of decreasing radius results in a cone Therefore, as t changes from 0 to 1, the transformed curve C1(u) sweeps out a surface. The curve C1(v) may be rotated and scaled, so for each t in the domain of curve C2(v), curve C1(u) is moved to the point C2(t), with possible rotation and/or scaling. The swept surface is the surface generated by moving curve C1(v) along curve C2(u). Swept Surfaces are created by two curves C1(u) and C2(v).The image below shows four possible types of continuity:įrom The Inventor Mentor: Programming Object-Oriented 3D Graphics with Open Inventor™, Release 2 Continuity is defined when the endpoints of two curves meet.What this means is that our visual system assumes that we are viewing something from a non-accidental point of view, and we believe what we see unless there is information to the contrary. External lines are viewed as the boundary of the shape.Acute and obtuse angles are interpreted as 90° angles in perspective.Continuous straight lines are interpreted as continuous straight lines. ![]() Two-dimensional parallel lines should be interpreted as three-dimensional parallel lines.Two-dimensional straight lines should be interpreted as three-dimensional straight lines.The Generic View Principle is our assumption that we are viewing images from a generic, or general, view. given the chance to interpret a drawing or image as three-dimensional, we will. As our visual system automatically assigns depth to each point in an image. The 2D drawing above is not impossible, but our three-dimensional interpretation of it is. A form is called impossible when our mind tries to interpret it as a three-dimensional object in Euclidean space, with straight edges and planar faces, instead of interpreting it as a two-dimensional object drawn on the paper plane.Ī family of impossible figures studied by knot theory, by Corinne Cerf ![]()
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