1. Know the spatial position of each point, and draw its projection map (the size is measured by the three-dimensional map and rounded up)
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2. Knowing one projection of the point and the following conditions, find the other two projections.
The distance between point A and surface V is 20mm.
(2) Point B is 10mm to the left of point A.
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3. Known point A (35, 20, 20),
B(15, 0, 25), ask for their projection diagrams.
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4. Knowing the two projections of each point, find the third projection.
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5. Determine the relative position of the following points.
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6. It is known that point B is 10mm to the left of point A, 15mm below and 10mm in front; point C is 10mm directly in front of point D, and the three-sided projection of point B and point C is made.
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7. Point A is known (10, 10, 15); point B is 20, 15, and 5 from the projected projection surface W, V, and H respectively; point C is 10 to the left of point A, 10 in front, and 5 above, and make A , B, C three-sided projection.
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8. It is known that the distances from point A to surfaces H and V are equal, so find a' and a". If the distances from point B to surfaces H, V, and W are equal, what is the relationship between the three coordinate values of point B? Each projection of B.
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9. Judge the relative positions of the following straight lines to the projection plane, and draw the three-plane projection.
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10. Make a line segment through point A so that it satisfies the following conditions (discussion: there are several solutions to the following questions, and only one solution is made).
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11. Find the real length of the line segment AB and its inclination angles α and β with the H and V planes
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Knowledge point: Calculate the inclination angle of a straight line and the real length of a line segment by the right triangle method.
1. Analysis: 1) According to the process of solving the inclination angle of the straight line and the projection surface and the actual length of the line segment by using the right triangle method, we can know that in a triangle whose length is two right-angled sides from the Z coordinate difference between the two ends of the line segment and the horizontal projection length of the line segment, The hypotenuse is equal to the real length of the line segment, and the angle between the hypotenuse and the right-angled side of the horizontal projection length is equal to α;
2) In a triangle whose length is right-angled by the Y coordinate difference between the two ends of the line segment and the length of the front projection, it can reflect the angle between the line segment and the V surface and the actual length of the line segment.
3) It can be seen from the projection diagram that the horizontal projection length and the front projection length of the line segment, and the Y coordinate difference and Z coordinate difference between the two ends of the line segment can be obtained by drawing.
2. Drawing steps: 1) Make horizontal lines through a' and b respectively, and the two straight lines intersect the line bb' and aa' respectively at point 1 and point 2;
2) Make the perpendicular line of a'b' through point a', and the perpendicular line of ab through point b; and intercept a'A1=a2(ΔYab), bb=b'1(ΔZab) respectively on the two perpendicular lines
3) Connect b′A1 and aB1 respectively with line segments; the result is shown in the figure.
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12. Mark the three-sided projection of the ridges AB, BC, and CD in the projection diagram of the object.
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13. Complete the three-plane projection of AB, and find a point K on AB, so that the distances from point K to planes H and V are equal.
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Knowledge points: the projection of a straight line; the correspondence between the distance from a point to the projection surface and the coordinates; the projection of a point on a straight line.
Analysis: 1) On the side: the OZ axis is the accumulation projection of the V surface, and the OYw axis is the accumulation projection of the H surface;
2) The distance from point K to surface H and surface V is equal, that is, ZK=YK, then point K must be on the plane that divides the angles of surfaces H and V, and the side projection of this surface is the angle of the axes OZ and OYW. Wire;
3) The intersection point k" between the bisecting angle line and a"b" is the equidistant point from AB to H and V planes.
The answer is shown in the picture below:
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14. Find the real length of the line segment CD and the angle β between it and the V surface.
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Knowledge points: the projection of a straight line, its real length and its angle with the projection surface.
Tips: 1) c′C1=c″1;
2）∠C1b′c′=β；
3) C1d' is the real length.
The answer is shown in the picture below:
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15. Find the real shape of ΔABC.
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Knowledge points: the projection and real length of a straight line; the real shape of a triangle.
1. Analysis: 1) It can be known from elementary geometry that a certain triangle can be drawn from two sides and their included angles, two angles and their included sides, or three sides (real length) of a known triangle. Now according to the horizontal and frontal projections of ΔABC, AC is a horizontal line, and its horizontal projection reflects the actual length of line segment AB, that is, ac=AC; similarly, a′b′=AB. As long as the actual length of BC is obtained, ΔABC can be made. 2) Use the method in exercise 1-11 to find the real length of BC. 3) Make ΔABC with the three sides of line segment ac, a'b' and b'C; ΔABC is what you want.
2. Drawing steps: 1) Draw a line parallel to the ox axis through point b, and this line intersects cc' at point 1; 2) Draw a straight line perpendicular to b'c' through c', and intercept c'C1= on this line b1; 3) Connect b′C1 with a line segment, b′C1 is the real length of side BC, that is, b′C1=BC; Draw an arc with a radius, and the two arcs intersect at point B; connect points a, B and point cB with a line segment, then ΔaBc≡ΔABC.
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16. Given the angle β=30° between the line segment AB and the V plane, find its horizontal projection.
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Knowledge point: Knowing a projection length of a straight line and an angle between it and the projection surface, use the right triangle method to obtain the difference of the third coordinate, so as to obtain other projections of the straight line.
In the drawing of solving the real length and inclination angle of a line segment with the right triangle method, the three sides of the right triangle containing β are respectively: the hypotenuse→the real length of the line segment, the right angle side of the adjacent side of the β angle→the length of the front projection of the line segment, Opposite side of β angle → Y coordinate difference between the two ends of the line segment. At this time, if the frontal projection of the line segment and its β angle are known, the problem is easy to solve.
The answer is shown in the picture below:
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17. Given the line segment EF=35mm, and its projections e′f′ and e″, find the projection of point K on EF, so that EK is the known length L.
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18. Given the line segment CD=45mm, find its front projection.
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Knowledge point: Use the right triangle method to find the projection of a straight line.
Since the real length of the line segment and its horizontal projection are known, in a right-angled triangle with the horizontal projection as the right-angled side and the real length of the line segment as the hypotenuse, the other right-angled side is the Z coordinate of the two ends of the frontal projection of the line segment Difference,
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19. Find a point C on the known line segment AB, make AC:CB=1:2, and find the projection of point C
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Knowledge point: Points belong to the properties of straight lines: the ratio between points and line segments is fixed. Note the application of the proportional triangle method to the straight line at a particular location.
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20. Through point A, make a line segment AB with a real length of 30mm. The included angles between it and H and V planes are α=45° and β=30° respectively; how many solutions are there for this problem?
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21. Find a point C (c, c′) on the straight line AB, so that the distance from point C to surface H is 15mm.
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22. Determine the relative position of the following straight lines.
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Knowledge point: the positional relationship of two straight lines—parallel, intersecting and interlacing.
1. Parallel conditions: a. Parallel projections on the same plane (including overlapping projections—coplanar); b. The proportional relationship remains unchanged; c. The directions are consistent.
2. Intersection condition: the intersection point is unique—the projected intersection point conforms to the property that the point belongs to a straight line.
3. Staggered conditions: non-parallel and intersecting.
Note: When it is a coplanar straight line, the positions of the two lines are only parallel and intersecting. Judgment at this time: just look at the other projection parallel—that is, the two lines are parallel (such as in 1: AB and CD); non-parallel—that is, they intersect (such as in 2: EF and GH; in 4: AB and EF).
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23. Make line segment AB through point A, make AB∥CD, and the actual length of AB is 30mm.
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Knowledge point: Use the right triangle method to find the projection of a straight line and the angle between it and the projection surface.
AB∥CD is ab∥cd, a′b′∥c′d′, a″b″∥cd″, therefore, a straight line al∥ab passes through point a, a′l′∥a′b′ passes through a′ Then: ab∥al, a′b′∥a′l′. The solution can be obtained by using the right triangle to find the real length of the line segment.
The answer is shown in the picture below:
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