Three-dimensional Plane DiagramsDate: 03/10/99 at 17:05:12 From: Lindsay Subject: Three-dimensional Plane Diagrams I am having trouble learning how to draw planes, and the special ways they are supposed to be drawn. For example, draw two parallel planes with another plane intersecting them, or draw two parallel planes with an intersecting line. Please help. Date: 03/11/99 at 12:48:48 From: Doctor Peterson Subject: Re: Three-dimensional Plane Diagrams Actually it is not too hard, as long as we do not try to get too fancy. Here is a horizontal plane: +----------------------------+ / / / / / / / / / / / / / / +----------------------------+ It looks horizontal because there's a horizontal line across the front. Here is a vertical plane turned so we see the right side, and another turned to show the left side: + + /| |\ / | | \ / | | \ / | | \ / | | \ + | | + | | | | | | | | | | | | | | | | | | | | | + + | | / \ | | / \ | | / \ | | / \ | |/ \| + + These look vertical because there's a vertical line along the front (or side). I can make them intersect just by putting them together at an edge: + /|\ / | \ / | \ / | \ / | \ + | + | | | | | | | | | | | | | | | | + | | / \ | | / \ | | / \ | | / \ | |/ \| + + Two parallel planes will look identical; here are two vertical planes intersected by a horizontal plane: + + /| /| / | / | / | / | / | / | +--/----+--------/----+--------+ / + /| + /| / / | / | | / | / / | / | | / | / / | / | | / | / / |/ | |/ | / +-------+-------------+--------+ | + | + | / | / | / | / | / | / | / | / |/ |/ + + I made the edges of the planes intersect so it is easy to see how they are related. (Of course, planes do not really have edges - this really pictures only part of the planes, so you can see something.) I also drew lines where the planes intersect, which join the points where the edges intersect. I chose to make the planes transparent; you can make them opaque by erasing the parts that are hidden behind planes: + + /| /| / | / | / | / | / | / | +--/ +--------/ +--------+ / + / + / / / | / | / / / | / | / / / | / | / / / |/ |/ / +-------+-------------+--------+ | + | + | / | / | / | / | / | / | / | / |/ |/ + + That is starting to get pretty impressive, isn't it! You could make the intersecting plane tilted so it is not perpendicular to the others, but I am not going to push my luck. For my last performance, I will give you your parallel planes with an intersecting line. To make it clear that the line is not parallel to the planes, I will send it in a different direction than the edges of the planes; to show where it intersects, I will make points, and make the planes opaque: \ + + \ /| /| / | / | / +| / | / \ / | / |\ / | + | \ + | | | \ | | | | \ | | | | \ | | | | \ | | | | \| | | + | + | / | / | / | + / | / | \ | / | / \ |/ |/ \ + + \ It really does not make much difference where you put those points of intersection, as long as they are on the line and within the plane. I hope that helps. If I have left out something, please write back. This is fun! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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