The 3D coordinate system is the established concept for defining an orientation in space.<\/p>\n
It describes the exact position of a point and how to find it in space. The 3D coordinate system is of great importance in general geometry, analysis and especially in vector calculus. <\/p>\n
Three-dimensional calculations are the basis for imaging, plastic processes and for milling programs in optical 3D measurement<\/a>.<\/p>\n Thanks to the increase in computing power, graphical and productive applications with the 3D coordinate system are now much cheaper than they were a few years ago.<\/p>\n <\/p>\n A 3D coordinate system is an extension of the 2D coordinate system<\/a>, which in turn is based on the number line.<\/p>\n It is used in mathematics and physics to represent ratios, trends and extreme values.<\/p>\n In applied mechanics, the coordinate system is used to graphically represent or simulate bodies.<\/p>\n The basis of the coordinate system is its axes. They are generally described as the X, Y and Z axes. However, these names are only placeholders that can be replaced by other designations depending on the application. <\/p>\n In a 2D coordinate system, the x-axis (abscissa) serves as the horizontal line and the y-axis (ordinate) as the vertical line. The Cartesian coordinate system, also known as the “axis cross”, is used to clearly assign values of rational functions. <\/p>\n In general algebra, the representation of surfaces is only achieved by drawing several functions in the same axis cross.<\/p>\n With the help of higher mathematics, more precisely differential functions and integrals, even complex contours can be represented exactly.<\/p>\n The contours can be intersected to form surfaces of any complexity. The axis cross defines four quadrants that can represent values in all positive and negative ranges. <\/p>\n In a 3D coordinate system, the axes are extended by the Z-axis and are not arranged in the same way as in the Cartesian axis cross: The X-axis now runs into the depth of space, the Y-axis becomes the horizontal and the Z-axis is the vertical.<\/p>\n In the 3D coordinate system, the quadrants are defined as cubic spaces, which can, however, be extended infinitely. In practical applications, however, only the purely positively doped quadrant plays a role in both the 2D and 3D coordinate systems. <\/p>\n A 2D<\/a> or 3D coordinate system is determined by its scaling. This can be the same for all axes, but does not necessarily have to be. <\/p>\n It may be useful to add a factor to the scaling of the individual axes. This makes it easier to display interesting sections of functions that are characterized by large extreme values, for example. <\/p>\n<\/div>\n In technical applications, the 3D coordinate system is used to capture, simulate and program CNC machines. In data acquisition, the 3D coordinate system is used as a reference frame for scanned objects, for example in industrial computer tomography<\/a>. <\/p>\n Photometry<\/a> or laser scanning systems can be used to capture any object in such a way that it can be transferred to a graphic display and evaluated there. The reverse engineering<\/a> process, also known as surface reconstruction, enables a complete view of the component without damaging it. <\/p>\n The simulation can also be created directly using design programs without having to scan a physical object first.<\/p>\n This is also usually the usual way. Emulators are used to program a CNC machine tool, for example a 5-axis milling machine. They can capture and convert any object so that it can be produced on a machine tool. <\/p>\n However, classic HEIDENHAIN or SIEMENS programming is still popular today for simple workpieces. Traditional CNC programming is based on vector calculation and is quite easy to implement. <\/p>\n Vector calculation allows precisely defined solids and positions to be represented with just a few entries. For example, it is sufficient to specify the spatial diagonal of a semi-finished block. <\/p>\n The 3D coordinate system of the CNC program, for example HEIDENHAIN, can already calculate the exact length, width and height of the basic block from this simple information.<\/p>\n All that is required are the coordinates of one of the upper corners. Example: A block with the same edge lengths is calculated from this, as the zero point 0\/0\/0 is already pre-programmed. The cycle is then precisely defined by specifying the coordinates and tools for each step. <\/p>\n In addition to subtractive processes, such as machining, adaptive processes are playing an increasingly important role.<\/p>\n The 3D printing systems developed from prototyping<\/a> are now so advanced that they are also being used for initial applications in series production.<\/p>\n The programming of adaptive systems is very similar to subtractive methods due to the use of a 3D coordinate system.<\/p>\n The 3D coordinate system also plays a major role in quality assurance. Here, the 3D coordinate measuring systems<\/a> are the standard application, especially for large and complex components<\/a>. Visual measurement reports<\/a> can be used to optimize and correct test specimens. <\/p>\n In these systems, the product to be measured or inspected is placed in the working area of a gantry robot.<\/p>\n This then guides a highly sensitive probe to all the designated points and thus ensures a valid target\/actual comparison.<\/p>\n The traditional scanning systems are quite precise, but they are also very slow and expensive. They are therefore increasingly being replaced by photographic processes or the so-called laser scans. <\/p>\n These non-contact methods are much faster and cheaper. In terms of precision, they already come close to traditional tactile methods<\/a>. <\/p>\n Today, users have a large selection of programs at their disposal for graphical processing or the creation of three-dimensional bodies.<\/p>\n 20 years ago, enormous computing power was still required to create plastic graphics. The programs and their products were still extremely expensive. <\/p>\n This has changed fundamentally with the increasing and cheaper computing power of modern computers.<\/p>\n In fact, there are already free programs available for users to download today that can be used to create very interesting graphics and illustrations.<\/p>\n The basic principle of these systems is always very similar: if you know your way around a 3D coordinate system, you can quickly implement these applications.<\/p>\nStructure of a coordinate system in space<\/h2>\n
The 3D coordinate system in technical applications<\/h2>\n
Traditional CNC programming<\/h2>\n
\nIf the program is given the coordinates 150\/250\/350, it knows that the end point of the basic block for the new workpiece is 150 millimetres high, 250 millimetres wide and 350 millimetres high. <\/p>\n3D coordinate systems in adaptive processes<\/h2>\n
3D coordinate systems in quality assurance<\/h2>\n
Programs for imaging three-dimensional bodies<\/h2>\n