All we will care about is its magnitude, and of course its sign (positive or negative)!ĭo this for each of the other 4 points around the closed path, and add up the values from this calculation at each vertex. This is a measure of whether the next segment after the vertex has bent to the left or right, and by how much. So, to get back to just a measure of the angle you need to divide this value, ( -16), by the product of the magnitudes of the two vectors. The magnitude of this value ( -16), is a measure of the sine of the angle between the 2 original vectors, multiplied by the product of the magnitudes of the 2 vectors.Īctually, another formula for its value isĪ X B (Cross Product) = |A| * |B| * sin(AB). The formula for calculating the magnitude of the k or z-axis component is Given that all cross-products produce a vector perpendicular to the plane of two vectors being multiplied, the determinant of the matrix above only has a k, (or z-axis) component. The third (zero)-valued coordinate is there because the cross product concept is a 3-D construct, and so we extend these 2-D vectors into 3-D in order to apply the cross-product: i j k These two edges are themselves vectors, whose x and y coordinates can be determined by subtracting the coordinates of their start and end points:ĮdgeE = point0 - point4 = (1, 0) - (5, 0) = (-4, 0) andĮdgeA = point1 - point0 = (6, 4) - (1, 0) = (5, 4) andĪnd the cross product of these two adjoining edges is calculated using the determinant of the following matrix, which is constructed by putting the coordinates of the two vectors below the symbols representing the three coordinate axis ( i, j, & k). So, for each vertex (point) of the polygon, calculate the cross-product magnitude of the two adjoining edges: Using your data:ĮdgeA is the segment from point0 to point1 and The magnitude of this vector is proportional to the sine of the angle between the two original edges, so it reaches a maximum when they are perpendicular, and tapers off to disappear when the edges are collinear (parallel). Then the cross product of two successive edges is a vector in the z-direction, (positive z-direction if the second segment is clockwise, minus z-direction if it's counter-clockwise). Imagine that each edge of your polygon is a vector in the x-y plane of a three-dimensional (3-D) xyz space. To see it move counterclockwise, click here.Īnd if you haven’t wasted enough time staring at the Necker cube and spinning dancer, check out these fun optical illusions.The cross product measures the degree of perpendicular-ness of two vectors. To see the lined image moving clockwise, click here. I did finally see the dancer flip, but it was only after using a sort of cheat sheet that draws a line on the dancer’s standing leg. Toppino says in people who can’t see the reversal, it may be that one underlying neural structure is more dominant, but once someone finally manages to see the flip, it will start I tried this several times,īut it never flipped. Toppino advises staring at one part of the image, such as the foot, and most of the time it will eventually flip. Sometimes, a person will stare at an image and it will never reverse. We can understand why it is these figures reverse then we’re in a position to understand something pretty fundamental to how the visual system contributes to the conscious experience.” Toppino, chair of the department of psychology at Villanova University. “What’s happening here to cause the flip is something happening entirely within the visual system,” said Thomas C. A moving rotating Necker cube can be seen here. It jumps to the back and the face of the cube shifts. The picture also lacks depth cues, so sometimes the face of the cube appears on the lower left, but sometimes Perhaps the most-studied reversible image is the Necker cube, which looks like the wire-frame of a cube. Most people, if they stare at the image long enough, will eventually see her turn both ways. Her as standing on her right leg and spinning to the left. As a result, your eyes will sometimes see the dancer standing on her left leg and spinning to the right. The silhouette image of the spinning dancer doesn’t have any depth cues. Images like this one have been long studied by scientists to learn more about how vision works. It is simply an optical illusion called a reversible, or ambiguous, image. Goes, you are using more of your right brain, and if you see it moving counterclockwise, you are more of a left-brained person.īut while the dancer does indeed reflect the brain savvy of its creator, Japanese Web designer Nobuyuki Kayahara, it is not a brain test. If you see the dancer spinning clockwise, the story A popular e-mail going around features a spinning dancer that has been touted as a test of whether you are right-brained and creative or left-brained and logical.
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