CoreXY is a technique used to move the printhead of a 3D printer or the toolhead in CNC machines in the horizontal plane. The advantage of this technique is that the two motors used to perform the movement in the horizontal plane are stationary and do not have to move themselves, which can result in less moving mass. Instead, drive belts are used which are connected in an intricate way [1] to provide movement in a Cartesian coordinate system.
For movement along the x-axis, both motors must rotate in the same direction. For movement along the y-axis, the motors must rotate in opposite directions. If only one motor rotates, the movement will be diagonal.
The movement can be mathematically described as follows. If is the movement of the first motor and the movement of the second motor, the movement in the x and y directions is given by:
Other Cartesian 3D printers which do not use the CoreXY technique most commonly also use two motors for the xy-plane, but where one motor is independently responsible for movement along the x-axis, and the other independently responsible for movement along the y-axis. This is sometimes called a Cartesian technique. [2] "Bed slinger" is a cartesian variant where the build surface moves along the y-axis, and the print head moves along the x-axis, and this technique is used on amongst other the Prusa i3 and clones.
A centripetal force is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In the theory of Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.
A Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines called coordinate line is called a coordinate axis or just axis of the system, and the point where they meet is the origin and has (0, 0) as coordinates.
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.
A gyrocompass is a type of non-magnetic compass which is based on a fast-spinning disc and the rotation of the Earth to find geographical direction automatically. The use of a gyrocompass is one of the seven fundamental ways to determine the heading of a vehicle. A gyroscope is an essential component of a gyrocompass, but they are different devices; a gyrocompass is built to use the effect of gyroscopic precession, which is a distinctive aspect of the general gyroscopic effect. Gyrocompasses are widely used for navigation on ships, because they have two significant advantages over magnetic compasses:
The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere, a primary direction towards the vernal equinox, and a right-handed convention.
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.
The stream function is defined for incompressible (divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in 1781. The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes.
In physics, Hooke's law is an empirical law which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.
In materials science and solid mechanics, Poisson's ratio (nu) is a measure of the Poisson effect, the deformation of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2–0.3. The ratio is named after the French mathematician and physicist Siméon Poisson.
In statistics, Deming regression, named after W. Edwards Deming, is an errors-in-variables model which tries to find the line of best fit for a two-dimensional dataset. It differs from the simple linear regression in that it accounts for errors in observations on both the x- and the y- axis. It is a special case of total least squares, which allows for any number of predictors and a more complicated error structure.
Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.
In continuum mechanics, the Cauchy stress tensor, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector e to the traction vector T(e) across an imaginary surface perpendicular to e:
In colorimetry the OSA-UCS is a color space first published in 1947 and developed by the Optical Society of America’s Committee on Uniform Color Scales. Previously created color order systems, such as the Munsell color system, failed to represent perceptual uniformity in all directions. The committee decided that, in order to accurately represent uniform color differences in each direction, a new shape of three dimensional Cartesian geometry would need to be used.
The instant center of rotation is the point fixed to a body undergoing planar movement that has zero velocity at a particular instant of time. At this instant, the velocity vectors of the other points in the body generate a circular field around this point which is identical to what is generated by a pure rotation.
In physics and chemistry, specifically in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and electron spin resonance (ESR), the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (Mx, My, Mz) as a function of time when relaxation times T1 and T2 are present. These are phenomenological equations that were introduced by Felix Bloch in 1946. Sometimes they are called the equations of motion of nuclear magnetization. They are analogous to the Maxwell–Bloch equations.
