I am trying to figure out a way to check if the plane intersects the pyramid. Computational complexity matters to me for this problem, so I have tried some various things to narrow down the number of possible planes:

Actually step 2 is done using weighted least square process. Each intersection line is assigned with a weight. Weight is proportional to the angle between normal vectors of corresponding planes. in this step, I tried to find the point which is closest to all the intersection lines i.e. point T. according to the weights, line positions might change with respect to the influence of high weight line. That mean, original planes could change little bit. So I want to show that these new positions of planes are well fitted for the original point data than original planes.


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That said, I will mention that the only difference between the planes you started with and the planes your procedure ends up with should be due to floating point error. This is because, geometrically speaking, all three lines should intersect at the same point as the planes that generated them.

In a pyramid I have the side plans $25^{\circ}$ from vertical line. I will make a L-shaped support in the corners. In the CAD I can see it will be $100.29^{\circ}$ between the planes. How can calculate this with formulas? Could not attach picture in the system but I can seen separate.

In this work we explored class separability in feature spaces built on extended representations of pixel planes (EPP) produced using scale pyramid, subband pyramid, and image transforms. The image transforms included Chebyshev, Fourier, wavelets, gradient and Laplacian; we also utilized transform combinations, including Fourier, Chebyshev and wavelets of the gradient transform, as well as Fourier of the Laplacian transform. We demonstrate that all three types of EPP promote class separation. We also explored the effect of EPP on suboptimal feature libraries, using only textural features in one case and only Haralick features in another. The effect of EPP was especially clear for these suboptimal libraries, where the transform-based representations were found to increase separability to a greater extent than scale or subband pyramids. EPP can be particularly useful in new applications where optimal features have not yet been developed.

There is only 1 unique 4 fold axis, because each is perpendicular to a similar looking face (the faces of the cube). There is only one unique 3-fold rotoinversion axes, because all of them stick out of the corners of the cube, and all are related by the 4-fold symmetry. And, there is only 1 unique 2-fold axis, because all of the others stick out of the edges of the cube and are related by the mirror planes the other set of 2-fold axes. So, we write a 4, a , and a 2 for each of the unique rotation axes. 4 2

Characterized by a single 4-fold or 4-fold rotoinversion axis.  Tetragonal-pyramidal Class, 4, Symmetry content - 1A4

 

 Since this class has a single 4-fold axis and no mirror planes, there are no pyramid faces on the bottom of the crystal. Wulfinite is the only mineral known to crystallize in this class.

 

 

 

    Tetragonal-disphenoidal Class, , Symmetry content - 14

 

 With only a single 4-fold rotoinversion axis, the disphenoid faces consist of two identical faces on top, and two identical faces on the bottom, offset by 90o. Note that there are no mirror planes in this class. Only one rare mineral is known to form crystals of this class.    Tetragonal-dipyramidal Class, 4/m, Symmetry content - 1A4, 1m, i

 

 This class has a single 4-fold axis perpendicular to a mirror plane. This results in 4 pyramid faces on top that are reflected across the mirror plane to form 4 identical faces on the bottom of the crystal. Scheelite and scapolite are the only common minerals in this class.

    Tetragonal-trapezohedral Class, 422, Symmetry content - 1A4, 4A2

 

 This class has a 4 fold axis perpendicular to 4 2-fold axes. There are no mirror planes. Only one rare mineral belongs to this class.   Ditetragonal-pyramidal Class, 4mm, Symmetry content - 1A4, 4m

 

 This class has a single 4-fold axis and 4 mirror planes. The mirror planes are not shown in the diagram, but would cut through the edges and center of the faces shown. Note that the ditetragonal pyramid is a set of 8 faces that form a pyramid on the top of the crystal. Only one rare mineral forms in the crystal class.

 

   Tetragonal-scalenohedral Class, 2m, Symmetry Content - 14, 2A2, 2m

 

 This class has a 4-fold rotoinversion axis that is perpendicular to 2 2-fold rotation axes. The 2 mirror planes a parallel to the and are at 45o to the 2-fold axes. Chalcopyrite and stannite are the only common minerals with crystals in this class.

 

    Ditetragonal-dipyramidal Class, 4/m2/m2/m, Symmetry content - 1A4, 4A2, 5m, i

 

 This class has the most symmetry of the tetragonal system. It has a single 4-fold axis that is perpendicular to 4 2-fold axes. All of the 2-fold axes are perpendicular to mirror planes. Another mirror plane is perpendicular to the 4-fold axis. The mirror planes are not shown in the diagram, but would cut through all of the vertical edges and through the center of the pyramid faces. The fifth mirror plane is the horizontal plane. Note the ditetragonal-dipyramid consists of the 8 pyramid faces on the top and the 8 pyramid faces on the bottom.  

Flying through the haze over the pyramids of Giza, an experimental solar-powered airplane arrived on Wednesday in Egypt as part of its globe-circling voyage. googletag.cmd.push(function() { googletag.display('div-gpt-ad-1449240174198-2'); }); The Solar Impulse 2 departed from the Seville airport in Spain on Monday and landed at the Cairo International Airport on Wednesday morning. This leg of the trip had been expected to last about 50 hours and 30 minutes.The aircraft, piloted by Swiss aviator Andre Borschberg, had arrived in Seville on June 23 after an unprecedented three-day flight across the Atlantic.The around-the-world voyage began in March 2015 in Abu Dhabi in the United Arab Emirates and is due to finish there in the coming weeks.The wings of Solar Impulse 2, which stretch wider than those of a Boeing 747, are equipped with 17,000 solar cells that power propellers and charge batteries. The plane runs on stored energy at night.Ideal flight speed is about 45.06 kph (28 mph), although that can double during the day when the sun's rays are strongest.The solar project, which is estimated to cost more than $100 million, began in 2002 to highlight the importance of renewable energy and the spirit of innovation. An experimental solar-powered airplane is seen after landing in Cairo, Egypt, Wednesday, July 13, 2016. The experimental aircraft, Solar Impulse 2, flew out of the Seville airport in Spain on Monday and landed in Cairo on Wednesday. (AP Photo/Mohamed Elraai)  An experimental solar-powered airplane is seen after landing in Cairo, Egypt, Wednesday, July 13, 2016. The experimental aircraft, Solar Impulse 2, flew out of the Seville airport in Spain on Monday and landed in Cairo on Wednesday. (AP Photo/Mohamed Elraai)  An experimental solar-powered airplane lands in Cairo, Egypt, Wednesday, July 13, 2016. The experimental aircraft, Solar Impulse 2, flew out of the Seville airport in Spain on Monday and landed in Cairo on Wednesday. (AP Photo/Mohamed Elraai)    2016 The Associated Press. All rights reserved.

The common version of the system involved joining an "airplane" by paying a "pilot" to become one of eight "passengers".[1][2] Passengers who started at the fourth step paid to join. Already on the airplane were four "flight attendants" who were a step ahead, and two "co-pilots" next in line behind the pilot. Once a pilot collected $12,000 from passengers to retire, the group split into two "airplanes", with each co-pilot becoming the pilot of the new airplane, taking half the participants and promoting everyone a level. Bringing in new passengers sped up everyone's progression towards retiring as a pilot.[1] However, the structure of the scheme results in a participant losing the entire payment unless 14 new participants join.[2]

The scheme had spread from New York to Texas to California then South Florida by early 1987,[2] with police raiding meetings in all four states,[3][1][2][4] and reports of more airplane schemes operating in Dallas.[4] In Miami-Dade County, Florida, at least one recruiting session was reported with 1,000 attendees.[2] Though common versions at the time required passengers to pay $1,500 to receive $12,000 as a pilot,[1][2][4] some airplanes were being run with $5,000 passengers and a $40,000 pilot payout.[2]

The inclined plane is one of the classical simple machines. As the name suggests, it is a flat surface held at an angle to the horizontal. By moving an object up an inclined plane rather than directly from one height to another, the amount of force required is reduced, at the expense of increasing the distance the object must travel. Some examples of inclined planes are a children's slide, a loading ramp, aircraft wings, windmills, and propeller blades.

Around 2600 B.C.E., inclined planes in the form of ramps were used, at least in part, to raise the blocks of stone that make up the Great Pyramid. Between 1900 and 1400 B.C.E., inclined planes might also have been used to elevate and place large stone crosspieces at Stonehenge.

Examples where "inclined planes" are to be found include ramps, sloping roads, hills, windshields, funnels, children's slides, water slides, and carpenter's planes. Other examples include aircraft wings, helicopter rotors, propellers (for aircraft and boats), windmills, water wheels, turbine blades, rotary fan blades, and machine screws.

"'Push it Up the 'Inclined Plane" is an original song from the Sid the Science Kid episode "My Slide". The song, sung by Susie, describes the process the Egyptians used to build the pyramids - by pushing giant blocks up inclined planes. be457b7860

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