Abstract:
The 3D visualization kit helps visualize concepts relating to points, surfaces, planes, curves, contours, and vectors in three dimensions. The kit is an inexpensive solution provided along with accompanying materials for its use that will allow students to effectively visualize concepts in three dimensions and aid in understanding important calculations in multivariable calculus eliminating the abstraction normally associated with concepts in three dimensions.

Description:
FEDERAL GRANTS 
     This research was supported, in part by the National Science Foundation through their DUE-995256 and DUE-0442365 grants. The Government has certain rights in this invention. 
    
    
     FIELD OF THE INVENTION 
     This invention relates generally to a visual aid, which may be used as a teaching aid and more particularly, to a visualization kit that has a plurality of elements that are interconnected to represent mathematical three-dimensional figures. 
     BACKGROUND OF THE INVENTION 
     Currently, there exists a lack of geometric visualization in the student&#39;s comprehension of three-dimensional concepts in multivariable calculus. Unfortunately, this in turn makes it difficult for the student to understand the basic calculations involved in math and engineering classes. For example, most students are confused about the signs of the first and second derivatives in various directions, when confronted with a picture of a surface on an exam. They are unable to determine the slope of the line between two points in 3D. Nor could they easily determine which integral is larger when given two surfaces, one clearly above the other. 
     Computer software has provided enormous aids to students and professors wishing to visualize concepts in three dimensions. However, there are many concepts where the two dimensional nature of a computer screen can limit the effectiveness of these packages; particularly if students have a weak geometric background. For example, directional derivatives require the tangent line to a surface in a direction associated with the xy plane. In three dimensions, a surface can be placed over the xy plane, the direction on the xy plane can be indicated and the concept can be visualized quite easily. However, visualizing a precise direction and its associated tangent line on a 2D computer screen is often difficult for students. Correspondingly, a more effective pedagogical approach is to use physical 3D manipulatives. These allow visualization and motivation of, concepts in a real three dimensional space. Particularly when students are first being introduced to multivariable functions, this often proves more effective than a projection of three dimensions onto a two dimensional computer screen. 
     Thus, what is needed, is a simple, hands-on 3D tool for use as aid in teaching these concepts. 
     SUMMARY OF THE INVENTION 
     The present invention advantageously helps students of science and engineering visualize concepts relating to points, surfaces, planes, curves, contours, and vectors in three dimensions. 
     According to an aspect of the invention, a 3D kit is provided to aid students understand important calculations in multivariable calculus eliminating the abstraction normally associated with concepts in three dimensions. 
     According to another aspect of the invention, an inexpensive kit is provided along with the accompanying materials for its use that will allow students to effectively visualize concepts in three dimensions. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Further features and advantages of the invention will become apparent from the following detailed description taken in conjunction with the accompanying figures showing illustrative embodiments of the invention, in which: 
         FIG. 1  shows a pegboard according to the present invention. 
         FIG. 2  shows an axial peg according to the present invention. 
         FIG. 3  shows a plurality of stacked axial, pegs defining axis Z according to the present invention. 
         FIG. 4  shows a peg inserting/magnetic arrangement according to the present invention. 
         FIG. 5  shows an antenna representing a vector according to the present invention. 
         FIG. 6  shows a plane representation according to the present invention. 
         FIG. 7  shows a plurality of flexible metallic elements representing curves according to the present invention. 
         FIG. 8  shows a plurality of magnetic marbles representing points according to the present invention. 
         FIG. 9  shows a hemisphere element according to the present invention. 
         FIG. 10  shows a paraboloid element according to the present invention. 
         FIG. 11(   a )-( d ) illustrates the steps for forming a point according to the present invention. 
         FIG. 12(   a )-( b ) illustrates the steps for forming a curve according to the present invention. 
         FIG. 13(   a )-( b ) illustrates the steps for forming a vector according to the present invention. 
         FIG. 14(   a )-( c ) illustrates the steps for forming a paraboloid according to the present invention. 
         FIG. 15(   a )-( c ) illustrates the steps for forming a hemisphere according to the present invention. 
         FIG. 16(   a )-( b ) illustrates the steps for forming a plane according to the present invention. 
     
    
    
     Throughout the figures, the same reference numbers and characters, unless otherwise stated, are used to denote like elements, components, portions or features of the illustrated embodiments. The subject invention will be described in detail in conjunction with the accompanying figures, in view of the illustrative embodiments. 
     DETAILED DESCRIPTION OF THE INVENTION 
     The basic interconnecting components of the 3D visualization kit are shown in  FIGS. 1-10 . These components are used interchangeably in order to form and represent points, surfaces, planes, curves, contours, and vectors in three dimensions as will be shown and explain later. 
       FIG. 1  shows the basic component of the system in the form of a flat surface pegboard  1  having on its top surface a plurality of holes  2 . These holes are evenly and symmetrically spaced in a matrix-like arrangement. The surface of said pegboard  1  represents a two-dimensional arrangement (i.e., X-Y).  FIG. 2  shows an axial peg  3  designed to have a reciprocal dual-engaging arrangement. As can be seen, an inserting protrusion  5  is positioned on a side of said axial peg  3  and longitudinally opposing a receiving hole  4  located on an opposing surface of said axial peg  3 . A similar arrangement is provided between the upper and lower side of said axial peg  3 . Horizontal lines can be formed using axial peg  3  by using the center holes as shown in  FIG. 2 . As will be illustrated later, the axial peg  3  is used to represent a third dimension in relation to the two-dimensional arrangement provided by said pegboard  1 . A plurality of axial pegs  3  can be stacked on top of each other to represent an extending dimensional axis (i.e., Z axis) by placing the inserting protrusion  5  of one axial peg  3  into the receiving hole  4  of another axial peg  3  as shown in  FIG. 3 . A dimensional unit is assigned to the height of each axial peg  3  to represent points and distances in a three-dimensional environment (i.e., 3 units in the direction of the Z axis=3 stacked axial pegs). 
     Pegs  7  are provided to interconnect the pegboard  1  and the rest of the components of said 3D visualization kit when creating a three-dimensional element. A hole  8  is provided on one end of said peg  7  and an inserting protrusion  9  is provided on the other end of said peg  7  as shown in  FIG. 4 . As will be explained in detail later, this arrangement allows stacking a plurality of pegs  7  similar to the stacking configuration of said axial peg  3 . A metallic peg  10  has a protrusion to be inserted inside any hole of the 3D kit so that when a magnet  11  is placed on its other end it provides the means for holding up points, vectors, curves and planes in space.  7   a  denotes a peg  7  having interconnected metallic peg  10  and magnet  11 . In the present invention, vectors are represented by antennas  12  having a telescopic component  13  on one end thereof, allowing to selectively extend the length of said antenna  12 . Metallic sheets  14  are provided to represent planes as shown in  FIG. 6 . In a preferred embodiment, pipe cleaners  15  as shown in  FIG. 7 , are provided to represent curves as will be explained later in detail. Alternatively, any metallic rod-like flexible element can be used. Three-dimensional points are represented in the 3D visualizing kit by metallic balls or marbles as shown in  FIG. 8 . 
     Hemispheres and paraboloids are represented by components  17  and  19  as shown in  FIGS. 9 and 10 . Insertion protrusions  18  and  20  are inserted in the holes of the 3D kit to secure said Hemispheres and paraboloids when forming three-dimensional figures. 
     In operation, the above-explained components are selectively used and positioned to form three-dimensional representations as will be explained in conjunction with  FIGS. 11-16 . 
       FIG. 11  illustrates the necessary steps to form a point defined by (X, Y, Z) using the 3D visualization kit of the present invention. First, a plurality of stacked axial pegs  3  are provided and inserted into said pegboard  1  to represent axis Z as shown in step (a). Then, the exact position X and Y on the pegboard  1  is located. If desired, the linear movement along these axes can be mark on said pegboard as shown in step (a), to help the students remember its position and to aid with any required calculation. After the X,Y position of the point has been located, a peg  7  is inserted into the hole representing the said X,Y position as shown in step (b). Additional pegs  7  might be stacked to indicate linear distance or height in the direction of axis Z. As illustrated in steps (c) and (d), the metallic peg  10  is positioned on top of the last peg  7  to receive the magnetic marble  16 . 
       FIG. 12  illustrates the necessary steps to form a curve that passes through points (X 1 , Y 1 , Z 1 ), (X 2 , Y 2 , Z 2 ) and (X 3 , Y 3 , Z 3 ) using the 3D visualization kit of the present invention. On step (a), axis Z and points (X 1 , Y 1 , Z 1 ), (X 2 , Y 2 , Z 2 ) and (X 3 , Y 3 , Z 3 ) are defined and installed as previously explained above. Each peg  7   a  must have installed on its top the metallic peg  10  with magnet  11 . Then, the flexible metallic component  15  is molded to the desired form of the curve and magnetically placed in contact with said pegs  7   a  as shown in step (b). 
       FIG. 13  illustrates the necessary steps to form a vector that goes from point (X 1 , Y 1 , Z 1 ) to point (X 2 , Y 2 , Z 2 ) using the 3D visualization kit of the present invention. On step (a), axis Z, points (X 1 , Y 1 , Z 1 ) and (X 2 , Y 2 , Z 2 ) are defined and installed as previously explained above. Each peg  7   a  must have installed on its top the metallic peg  10  with magnet  11 . Then, one end of the antenna  12  is magnetically placed on top of the peg  7   a  defining point (X 1 , Y 1 , Z 1 ) and portion  13  is extended so that the other end of the antenna  12  magnetically rests on top of the peg  7   a  defining point (X 2 , Y 2 , Z 2 ) as shown in step (b). 
       FIGS. 14 and 15  illustrate the necessary steps to form a hemisphere and paraboloid with vertex in the origin (X 1 , Y 1 , Z 1 ) using the 3D visualization kit of the present invention. On step (a), points (X 1 , Y 1 , Z 1 ) to (X 4 , Y 4 , Z 4 ) are defined and marked on said pegboard  1 . Then, a peg  7  is inserted on each identified point to receive insertion protrusions  18  and  20  so that four hemisphere and paraboloid components  17  and  19  are positioned together to form the desired configuration as shown in steps (b) and (c). It should be noted that the procedures to form a hemisphere and paraboloid using the 3D visualization kit of the present invention is similar. 
       FIG. 16  illustrates the necessary steps to form a plane that passes through points (X 1 , Y 1 , Z 1 ), (X 2 , Y 2 , Z 2 ) and (X 3 , Y 3 , Z 3 ) using the 3D visualization kit of the present invention. On step (a), axis Z and points (X 1 , Y 1 , Z 1 ), (X 2 , Y 2 , Z 2 ) and (X 3 , Y 3 , Z 3 ) are defined and installed as previously explained above. Each peg  7   a  must have installed on its top the metallic peg  10  with magnet  11 . Then, the metallic sheet plane  14  is magnetically placed over pegs  7   a  as shown in step (b). 
     It is important to note that a water-based marker can be used to write on all materials of the 3D kit. This allows supplementing demonstrations in three dimensions with appropriate symbols, calculations, and numbers. 
     Although the invention has been described in conjunction with specific embodiments, it is evident that many alternatives and variations will be apparent to those skilled in the art in light of the foregoing description. Accordingly, the invention is intended to embrace all of the alternatives and variations that fall within the spirit and scope of the appended claims.