miércoles, 9 de mayo de 2012
Foundation
The dihedral system is a method of representation of multiple projections, where the elements are defined by their orthogonal projections on at least two planes of projection.
projection planes that we use are generally 3: plan, elevation and profile . Once they have been projected onto each of them the orthogonal views of the object are rotated until they match the three in the same plane. Figure cylinder projecting a point by point on the horizontal plane and the vertical PH PV. As is done by perpendicular, the circumference of the base becomes the elevation in a straight line, as the plane that contains perpendicular to the vertical. For be parallel to the floor, the upper face of the cylinder on this plane is transformed Like a circle. The line of intersection of vertical and horizontal plane is called land line .
Then turn the vertical plane 90 degrees until it coincides with the horizontal axis of rotation taking as the ground line. The rotationcauses the two views are perfectly aligned in lines orthogonal to the ground line.
The two views dihedral (in plan and elevation) would in this way. After the rotation of the object dihedral projections are always correlated.
We removed the reference contour of the two planes and we already have the plan and elevation. The earth line is represented by a straight dihedral system which separates the ground and with two elevation of the segments at their ends.
If the part has more complexity can be necessary to represent another view in a projection plane. In the figure we see the plane of the profile (PP).
In the figure we see in yellow what is projected in the vertical plane, in red and green on the green horizontal and the plane of the profile.
In the plant is placed orthogonal projection of the piece "view" from above (in yellow and light blue). Correlative to the above lists the elevation is the front view (orange, green and dark blue) and finally the profile of the (pink), as its name. The parties are not visible to the observer as part placed above to see the plant, or front to see the elevation are discontinuous. Dotted lines are lines that exist that are intersection of surfaces but which can not be appreciated by being behind a face.
A piece with their projections in plan elevation and profile, and a representation axonometricthereof. The piece, and has a prismatic hole whose edges are tangent to some faces of the figure, these lines appear as a continuous arrangement are visible.
We will represent the dihedral a prism system with another hollow prism at an angle in its interior for this purpose we have two procedures:
On the one hand we have the European system in which the observer is placed on top of the piece to see it from above ground. Is placed behind the piece to see it in the elevation or profile. With approach, which viewed from a distant point of view to avoid distortion of concurrent lines of perspective, it looks like the orthogonal projections of the object on the three planes of projection.
Once you have designed the three views on the three planes of projection, the profile plane is rotated 90 ° until it coincides with the plane of elevation. Then the two planes are rotated until they match the plant, all coinciding with the picture plane or paper. Rotation is always relative to the line of intersection of the planes.
On the other hand we have the American system, in which a practical representation of the same rule to the observer, the object is just as if it were reflected on the three planes of projection being these mirrors. The image of the three faces of the workpiece are as shown reflected on the three planes of projection.
To obtain the elevation and profile views, rotated 90 ° to both planes which coincide with the plane of the plant, in this way represent the three dihedral system views. The LT is a line segment formed by a great alternative to two small and so on.
In the figure we see a piece in isometric axonometric projections with three dihedral, plan, elevation, and profile. Like many faces of the figure are coincident with each other or with the continuation of other, sights dihedral is difficult to deduce the shape of the piece.
If we represent new views and show the inside of the piece by dashed lines facilitate the understanding of it. We have however the same problem as in the previous drawing, some edges of the piece are matching so you have very bad performance, not even showing their faces in different colors can be seen easily. As the axonometric isometric matches the edges confusing interpretation of the piece, should make a turn in the same space for a new axonometric projection, as in the diagram below.
In this view it is easier to interpret the piece, however as dashed lines are missing we can not interpret well the parts of the drawing is not visible from the point of view of the imaginary part (imaginary because it does not have the axonometric view, is an orthogonal projection cylindrical, hence all parallel edges of the piece also go parallel in the drawing, so it is said that parallelism is a projective invariant in this perspective).
In this new Trimetric axonometric projection, we can now differentiate with vivid clarity the various elements of the piece. Consequently, when views are not enough dihedral for the interpretation of the drawing, nor would this piece in different cuts and auxiliary views of the same-we must draw an axonometric perspective in which preferably the edges of the part coincide.
Quadrant in s. dihedral On the left we see the four quadrants of the dihedral system. In the first quadrant point A has two orthogonal projections A1 A2 on the horizontal and vertical respectively. By rotating the vertical plane in a counter clockwise, the vertical projection A2 point A is on the ground line while the horizontal A1 falls below it, as shown in the drawing to the right. In the second quadrant B with both horizontal and vertical projections B1 B2, respectively, are transformed by rotating the vertical plane B1, B2, both on the ground line, as shown in the drawing to the right. In the third quadrant C, we can observe that by rotating the vertical plane, the vertical projection of point C2 becomes below the earth line while the projection onto the horizontal plane C1 lies above the ground line, following the drawing on the right. a point D in the fourth quadrant with two horizontal and vertical projections D1 D2, respectively, we have by rotating the vertical plane 2 are transformed into points on a vertically aligned below the ground line, as shown the drawing of the right.
Representation of dihedral system elements by coordinates.
We have the two planes of projection, the horizontal and vertical. The intersection of two planes or ground line is considered the X axis (in red).The line perpendicular to the axis on the horizontal plane through any point is considered the axis (magenta). From this point or origin of coordinates (0.0) we make a vertical line and consider the axis Z.
In this way represent the points in the different quadrants: the point A is 20 units from the origin on the axis X., from this point to the right has a distance of five units (taken on the y axis) and height or altitude of three units (taken on the Z axis), so the coordinates of point A are (20 , 5, 3). In the opposite direction of that we used would have units with a negative value, for example, point B has coordinate (9, -7.4), this means that on the axis X. is nine units, which from this point to the left along the Y axis is -7 units, and from this point at a height or altitude of four units located at point B. A point which is in the second quadrant on the shaft and has a negative value and on the axis Z. its positive value, whereas on the axis X. can have either positive or negative.
Point C of the third quadrant is negative coordinates of both the axis and as the Z axis, so is the point coordinates (0, -8, -2), this means that has by restraining eight units and height dimension or two, both for being negative in the third quadrant.
The coordinate point is D (15.6, -20), in this quadrant Z always has a negative value, while the value of And it is always positive.
Dihedral system can facilitate understanding of the foundations of other systems of representation. Figure we have a cube that is projected at an angle and with parallel lines on a plane called the PC box. At least one of the faces of the cube is parallel to the picture plane. We note that a 10 cm vertical segment is transformed into a 9 cm, this means that the reduction is applied in this system of representation is nine tenths . We also have from the X axis begins to have an angle in the direction clockwise to orient the axis Y. The angle between these two axes is the angle of the isometric, 315 ° in this case. In the figure we can observe the representation dihedral of isometric, the projection in plan and elevation of each of the elements by parallel lines which transform Figure on the ground in isometric. On the ground will make the isometric of the figure, which is but the shadow of the figure projected on the floor plan. As shown in the raised red face of the cube becomes the XY plane on the ground, while the upper face of the cube (green) turns into perspective in a green face. http://perspectiva-caballera. blogspot.com /
In the figure we can see the foundation of the system axonometric three Cartesian axes XYZ is projected onto a plane by lines parallel and orthogonal to it. These three areas are the corner of a cube (together form two to two 90 °, which in geometry is called a trihedral trirrectángulo).
To get the true measure of the cube that is projected orthogonally on the picture plane, is fold down the sides, so that we can work on paper, on the picture plane PC.
On the face downcast (x) (y) are placed the views of the part and are projected orthogonally on the trace of the face downcast until cut to the axes x 'and', thus obtaining the reduced dimension on the same edges of the cube. In the drawing we have the representation of the dihedral trihedral system that has befallen one side, we observe the lowering in elevation by rotating the xy face. Folded over the face (x) (y) is placed one side of the figure and is projected orthogonal to the hinge means until it intercepts the xy axes.By projecting the flat shape have to influence the vertices of this axle of the axonometric xy, with what we have and the perspective of the face of the figure with its corresponding reduction axonometric and perspective. http://perspectiva-axonometrica.blogspot.com/
In the figure have the merits of the conical perspective, of the point of view V are parallel to the direction of the edges of the piece V-F1-F2 V until they intercept on the picture plane to the vanishing points F1 F2.N sides of the projection of the piece on the plane of the base extends into that cut to the picture plane, obtaining the traces tn to unite with the vanishing points F1. The intersection of these lines is the ring in perspective in yellow on the picture plane. As an example we have the prospect of n is d.
To obtain the heights of the piece is placed from the trace tn vertical dimensions is true extent and d projected to vanishing points F1 getting the perspective of these vertical lines . As we can see each point of the piece and its outlook U U 'are aligned with the view V, and this is the foundation of the conical perspective. In figure we have the representation of the dihedral system with two point perspective projections overlap. As shown in plan on the green square extend until n sides intersect the picture plane in the traces tn, got on the points raised above the ground line. Through the point of view V lines are parallel to the sides of the figure obtained in the base at the intersection with the picture plane in plant leak points F1 F2. These leakage points are projected onto the floor elevation on the line is up to the point of view, which is the F1 F2 horizon. Joining the vanishing points with the traces of the piece tn-F1 we have the prospect the green square on the elevation, which is the yellow square. From one of the traces of these lines are placed tn is the vertical measurements of the elements represent. These two measures are projected to the vanishing point where F1 and the vertical intercept of each point q from the perspective of the ring we have the prospect yellow conical shape with its vertical dimensions in perspective. We can see in the perspective projection of the point of view on the horizon, we call the main point, is such that lines each point with your perspective, for example you can see that a vertex of the figure in the elevation and U perspective U 'are aligned with the projection of the view V' on the picture plane in the elevation http://la-perspectiva-conica.blogspot.com/
http://proyeccion-gnomonica.blogspot.com/
http://proyeccion-central-conica.blogspot.com/
projection planes that we use are generally 3: plan, elevation and profile . Once they have been projected onto each of them the orthogonal views of the object are rotated until they match the three in the same plane. Figure cylinder projecting a point by point on the horizontal plane and the vertical PH PV. As is done by perpendicular, the circumference of the base becomes the elevation in a straight line, as the plane that contains perpendicular to the vertical. For be parallel to the floor, the upper face of the cylinder on this plane is transformed Like a circle. The line of intersection of vertical and horizontal plane is called land line .
Then turn the vertical plane 90 degrees until it coincides with the horizontal axis of rotation taking as the ground line. The rotationcauses the two views are perfectly aligned in lines orthogonal to the ground line.
The two views dihedral (in plan and elevation) would in this way. After the rotation of the object dihedral projections are always correlated.
We removed the reference contour of the two planes and we already have the plan and elevation. The earth line is represented by a straight dihedral system which separates the ground and with two elevation of the segments at their ends.
If the part has more complexity can be necessary to represent another view in a projection plane. In the figure we see the plane of the profile (PP).
In the figure we see in yellow what is projected in the vertical plane, in red and green on the green horizontal and the plane of the profile.
In the plant is placed orthogonal projection of the piece "view" from above (in yellow and light blue). Correlative to the above lists the elevation is the front view (orange, green and dark blue) and finally the profile of the (pink), as its name. The parties are not visible to the observer as part placed above to see the plant, or front to see the elevation are discontinuous. Dotted lines are lines that exist that are intersection of surfaces but which can not be appreciated by being behind a face.
A piece with their projections in plan elevation and profile, and a representation axonometricthereof. The piece, and has a prismatic hole whose edges are tangent to some faces of the figure, these lines appear as a continuous arrangement are visible.
We will represent the dihedral a prism system with another hollow prism at an angle in its interior for this purpose we have two procedures:
On the one hand we have the European system in which the observer is placed on top of the piece to see it from above ground. Is placed behind the piece to see it in the elevation or profile. With approach, which viewed from a distant point of view to avoid distortion of concurrent lines of perspective, it looks like the orthogonal projections of the object on the three planes of projection.
Once you have designed the three views on the three planes of projection, the profile plane is rotated 90 ° until it coincides with the plane of elevation. Then the two planes are rotated until they match the plant, all coinciding with the picture plane or paper. Rotation is always relative to the line of intersection of the planes.
On the other hand we have the American system, in which a practical representation of the same rule to the observer, the object is just as if it were reflected on the three planes of projection being these mirrors. The image of the three faces of the workpiece are as shown reflected on the three planes of projection.
To obtain the elevation and profile views, rotated 90 ° to both planes which coincide with the plane of the plant, in this way represent the three dihedral system views. The LT is a line segment formed by a great alternative to two small and so on.
In the figure we see a piece in isometric axonometric projections with three dihedral, plan, elevation, and profile. Like many faces of the figure are coincident with each other or with the continuation of other, sights dihedral is difficult to deduce the shape of the piece.
If we represent new views and show the inside of the piece by dashed lines facilitate the understanding of it. We have however the same problem as in the previous drawing, some edges of the piece are matching so you have very bad performance, not even showing their faces in different colors can be seen easily. As the axonometric isometric matches the edges confusing interpretation of the piece, should make a turn in the same space for a new axonometric projection, as in the diagram below.
In this view it is easier to interpret the piece, however as dashed lines are missing we can not interpret well the parts of the drawing is not visible from the point of view of the imaginary part (imaginary because it does not have the axonometric view, is an orthogonal projection cylindrical, hence all parallel edges of the piece also go parallel in the drawing, so it is said that parallelism is a projective invariant in this perspective).
In this new Trimetric axonometric projection, we can now differentiate with vivid clarity the various elements of the piece. Consequently, when views are not enough dihedral for the interpretation of the drawing, nor would this piece in different cuts and auxiliary views of the same-we must draw an axonometric perspective in which preferably the edges of the part coincide.
Quadrant in s. dihedral On the left we see the four quadrants of the dihedral system. In the first quadrant point A has two orthogonal projections A1 A2 on the horizontal and vertical respectively. By rotating the vertical plane in a counter clockwise, the vertical projection A2 point A is on the ground line while the horizontal A1 falls below it, as shown in the drawing to the right. In the second quadrant B with both horizontal and vertical projections B1 B2, respectively, are transformed by rotating the vertical plane B1, B2, both on the ground line, as shown in the drawing to the right. In the third quadrant C, we can observe that by rotating the vertical plane, the vertical projection of point C2 becomes below the earth line while the projection onto the horizontal plane C1 lies above the ground line, following the drawing on the right. a point D in the fourth quadrant with two horizontal and vertical projections D1 D2, respectively, we have by rotating the vertical plane 2 are transformed into points on a vertically aligned below the ground line, as shown the drawing of the right.
Representation of dihedral system elements by coordinates.
We have the two planes of projection, the horizontal and vertical. The intersection of two planes or ground line is considered the X axis (in red).The line perpendicular to the axis on the horizontal plane through any point is considered the axis (magenta). From this point or origin of coordinates (0.0) we make a vertical line and consider the axis Z.
In this way represent the points in the different quadrants: the point A is 20 units from the origin on the axis X., from this point to the right has a distance of five units (taken on the y axis) and height or altitude of three units (taken on the Z axis), so the coordinates of point A are (20 , 5, 3). In the opposite direction of that we used would have units with a negative value, for example, point B has coordinate (9, -7.4), this means that on the axis X. is nine units, which from this point to the left along the Y axis is -7 units, and from this point at a height or altitude of four units located at point B. A point which is in the second quadrant on the shaft and has a negative value and on the axis Z. its positive value, whereas on the axis X. can have either positive or negative.
Point C of the third quadrant is negative coordinates of both the axis and as the Z axis, so is the point coordinates (0, -8, -2), this means that has by restraining eight units and height dimension or two, both for being negative in the third quadrant.
The coordinate point is D (15.6, -20), in this quadrant Z always has a negative value, while the value of And it is always positive.
Dihedral system can facilitate understanding of the foundations of other systems of representation. Figure we have a cube that is projected at an angle and with parallel lines on a plane called the PC box. At least one of the faces of the cube is parallel to the picture plane. We note that a 10 cm vertical segment is transformed into a 9 cm, this means that the reduction is applied in this system of representation is nine tenths . We also have from the X axis begins to have an angle in the direction clockwise to orient the axis Y. The angle between these two axes is the angle of the isometric, 315 ° in this case. In the figure we can observe the representation dihedral of isometric, the projection in plan and elevation of each of the elements by parallel lines which transform Figure on the ground in isometric. On the ground will make the isometric of the figure, which is but the shadow of the figure projected on the floor plan. As shown in the raised red face of the cube becomes the XY plane on the ground, while the upper face of the cube (green) turns into perspective in a green face. http://perspectiva-caballera. blogspot.com /
In the figure we can see the foundation of the system axonometric three Cartesian axes XYZ is projected onto a plane by lines parallel and orthogonal to it. These three areas are the corner of a cube (together form two to two 90 °, which in geometry is called a trihedral trirrectángulo).
To get the true measure of the cube that is projected orthogonally on the picture plane, is fold down the sides, so that we can work on paper, on the picture plane PC.
On the face downcast (x) (y) are placed the views of the part and are projected orthogonally on the trace of the face downcast until cut to the axes x 'and', thus obtaining the reduced dimension on the same edges of the cube. In the drawing we have the representation of the dihedral trihedral system that has befallen one side, we observe the lowering in elevation by rotating the xy face. Folded over the face (x) (y) is placed one side of the figure and is projected orthogonal to the hinge means until it intercepts the xy axes.By projecting the flat shape have to influence the vertices of this axle of the axonometric xy, with what we have and the perspective of the face of the figure with its corresponding reduction axonometric and perspective. http://perspectiva-axonometrica.blogspot.com/
In the figure have the merits of the conical perspective, of the point of view V are parallel to the direction of the edges of the piece V-F1-F2 V until they intercept on the picture plane to the vanishing points F1 F2.N sides of the projection of the piece on the plane of the base extends into that cut to the picture plane, obtaining the traces tn to unite with the vanishing points F1. The intersection of these lines is the ring in perspective in yellow on the picture plane. As an example we have the prospect of n is d.
To obtain the heights of the piece is placed from the trace tn vertical dimensions is true extent and d projected to vanishing points F1 getting the perspective of these vertical lines . As we can see each point of the piece and its outlook U U 'are aligned with the view V, and this is the foundation of the conical perspective. In figure we have the representation of the dihedral system with two point perspective projections overlap. As shown in plan on the green square extend until n sides intersect the picture plane in the traces tn, got on the points raised above the ground line. Through the point of view V lines are parallel to the sides of the figure obtained in the base at the intersection with the picture plane in plant leak points F1 F2. These leakage points are projected onto the floor elevation on the line is up to the point of view, which is the F1 F2 horizon. Joining the vanishing points with the traces of the piece tn-F1 we have the prospect the green square on the elevation, which is the yellow square. From one of the traces of these lines are placed tn is the vertical measurements of the elements represent. These two measures are projected to the vanishing point where F1 and the vertical intercept of each point q from the perspective of the ring we have the prospect yellow conical shape with its vertical dimensions in perspective. We can see in the perspective projection of the point of view on the horizon, we call the main point, is such that lines each point with your perspective, for example you can see that a vertex of the figure in the elevation and U perspective U 'are aligned with the projection of the view V' on the picture plane in the elevation http://la-perspectiva-conica.blogspot.com/
http://proyeccion-gnomonica.blogspot.com/
http://proyeccion-central-conica.blogspot.com/
http://proyeccion-central-dinamica.blogspot.com.es/
In the figure we can observe the foundation for a perspective conic shape with a picture plane oblique with respect to the three sides of the same.
On the point of view are straight lines parallel to each of the edges of the figure to which intercept to the picture plane in vanishing points FM L. Aligning the point of view V with each of the vertexes of the figure at the intersection Y have the same perspective of each of the points Y '. In the figure we can observe the representation in plan elevation and profile of Figure above. As the picture plane coincides with the plane of drawing in this time have tended to tilt the plane of the base and the figure, which has at least one of its faces parallel to it. As noted in the plan and profile, we align the point of view with each of the points of the figure and at the intersection with the picture plane orthogonal do the same. For example, in aligning the point Y3 profile with the point of view we V3 J3, and aligning the corresponding plant Y1 to obtain V1 viewpoint J1. The intersection of the orthogonal in plan and elevation so the points of intersection Y1 J3 we determine the perspective of the figure in the sum J2, just as we proceed with the remaining points of the figure. We can see that this perspective is true, that figure is seen as a subject from the point of view V1 placed at that distance on the ground the plane of the PC box, if we extend the edges of the figure in perspective we note that cut into three vanishing points F2 L2 M2 coincide on the intersection of orthogonal passing through the vanishing points.As an example we obtain L2 orthogonal intersection of the points L1 and L3 on the ground in the profile. We can also observe that the point of view projected onto the picture plane in the elevation, which is the main point P2 is aligned with each point of the figure and its perspective, as an example the points Y2-J2-P2 are aligned. http :/ / perspective-of-box-inclinado.blogspot.com /
In the figure we can observe the foundation for a perspective conic shape with a picture plane oblique with respect to the three sides of the same.
On the point of view are straight lines parallel to each of the edges of the figure to which intercept to the picture plane in vanishing points FM L. Aligning the point of view V with each of the vertexes of the figure at the intersection Y have the same perspective of each of the points Y '. In the figure we can observe the representation in plan elevation and profile of Figure above. As the picture plane coincides with the plane of drawing in this time have tended to tilt the plane of the base and the figure, which has at least one of its faces parallel to it. As noted in the plan and profile, we align the point of view with each of the points of the figure and at the intersection with the picture plane orthogonal do the same. For example, in aligning the point Y3 profile with the point of view we V3 J3, and aligning the corresponding plant Y1 to obtain V1 viewpoint J1. The intersection of the orthogonal in plan and elevation so the points of intersection Y1 J3 we determine the perspective of the figure in the sum J2, just as we proceed with the remaining points of the figure. We can see that this perspective is true, that figure is seen as a subject from the point of view V1 placed at that distance on the ground the plane of the PC box, if we extend the edges of the figure in perspective we note that cut into three vanishing points F2 L2 M2 coincide on the intersection of orthogonal passing through the vanishing points.As an example we obtain L2 orthogonal intersection of the points L1 and L3 on the ground in the profile. We can also observe that the point of view projected onto the picture plane in the elevation, which is the main point P2 is aligned with each point of the figure and its perspective, as an example the points Y2-J2-P2 are aligned. http :/ / perspective-of-box-inclinado.blogspot.com /
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