MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C3EB7A.BC4FA6E0" This document is a Web archive file. If you are seeing this message, this means your browser or editor doesn't support Web archive files. For more information on the Web archive format, go to http://officeupdate.microsoft.com/office/webarchive.htm ------=_NextPart_01C3EB7A.BC4FA6E0 Content-Location: file:///C:/A59B3899/Ingenernay_grafika-Krasovskay.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1251" ИНЖЕНЕРНАЯ ГРАФИКА

Министерство путей сообщения Российской Федерации

Дальневосточный государственный университет путей сообщения

&nb= sp;

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Каф= едра “Системы автоматизированного
проектирования”

Т.С. Красовская

 

ИНЖЕНЕРНАЯ ГРАФИКА

Методические указания изучения дисциплины
и учебно-методический материал для самостоятельной работы
=

 

Хабаровск

2002

Рецензент Кандидат физико-математических наук, заведующий кафедрой “Математические и естественнонаучные дисциплины” Хабаровского военного института Федеральной пограничной службы Российской Федерации А.И. Ивлева

К 784

Красовская Т.С.<= o:p>

Инженерная графика: Методические указания / Т.С. Красовская. – Хабаровск: Изд-во ДВГУПС, 2002. – 47 с.

Методические указания содержат краткий теоретический материал для изучения дисциплины, а также пр= актикум и учебно-методический материал для самостоятельной работы и самоконтроля студента.

П= редназначены для студентов первого курса специальностей 330100, 220300, 071702.

ã Издательство Дальневосточного государственного университ= ета путей сообщения (ДВГУПС), 2002

 


 

Содержание

ВВЕДЕНИЕ

1. Практикум по инженерной графике

2. ЭТАЛОНЫ

Список литературы


 

ВВЕДЕНИЕ

1. Виды проецирования=

Основными видами проецирования являются центральное и параллельное проецирование.=

Центральное проецирование представляет собой общий случай проецирования геометрических образов из некоторого центр= а на данную плоскость (картину).

Пусть задана плоскость проекций П' (плоскость картины) и кри= вая линия k с точками А,= В и С (черт. 1). Возьмем некоторую точку S, не лежащу= ю в плоскости П'. Через = точку S и точки А, В и С кривой k проведем прямые (проецирующие лучи) до пересечения с плоскостью П' в точках А', В', C'. Проведя таким образом через = S и каждую точку кривой k прямые, получим в плоск= ости П' изображение k' кривой k.

В соответствии с полу= ченным построением введем следующие, понятия: точка S – центр (полюс) проекций, П' – плоскость проекций (плоскость картины); кривая k с принадлежащими = ей точками А, В, С – проецируемый объект (образ, элемент); точки А', В', С' – центральные проекции точек А, В, С и кривая k' – центральная проекция кривой k.

Рассматривая каждую пространственную фигуру как совокупность (множество) точек, можно сказать, = что проекция фигуры представляет собой множество проекций ее точек.

Совокупность прямых, проходящих через центр проекций и все точки проецируемого объекта (k), представляет собой, в общем случае, коническую= поверхность, поэтому этот вид проецирования иногда называют коническим.

Прямые и плоскости, проходящие через центр проекций, называются проецирующими.

Центральная проекция, обладая большой наглядностью, имеет существенные недостатки, заключающиеся в сложности построения изображения предмета и определения истинных размеров. Поэтому этот способ имеет огранич= енное применение в практике.

Центральное проецирование применяют при построении перспектив зданий и сооружений, в живописи и т. п.

Наглядность центра= льных проекций объясняется устройством зрительного аппарата человеческого глаза, который работает по принципу центрального проецирования. Так, например, оптический центр хрусталика глаза можно принять за центр проецирования (ч= ерт. 2), а сетчатку глаза за плоскость проекций. Луч зрения (проектирующий луч= ), идущий от видимого предмета, проходит через хрусталик, попадает на сетчат= ку глаза и дает изображение.

Параллельное проецирование можно рассматривать как частный случай центрального проецирования с бесконечно удаленным центром проекций. Осуществляется оно пучком параллельных прямых да= нного направления.

Пусть требуется построить параллельную проекцию кривой <= span class=3DSpellE>k на плоскость П' (черт. 3).

  Спроецируем в направлении S все= точки кривой k на плоскость П'. Чтобы спроецировать точки указанной криво= й, например А, В, С, нужно провести через них прямые, параллельные направлению 5 до пересечения с плоскостью П'. Точки пересечения А', В', С' проец= ирующих лучей с плоскостью П' и будут параллельными проекциями точек А, В и С. Таким образом, = можно построить проекции множества точек объекта, например кривой k. Совокупность проецирующих лучей в данном = случае образует цилиндрическую поверхность общего вида. Поэтому такой вид проецирования иногда называют цилиндрическим.

Если в качестве проецируемого объекта будет прямая а, то ее проекция в общем случае будет прямая а' (черт. 3). Совокупность проецирующих лучей здесь принадлежит плоскости СDD'С'. Такую плоскость принято называть проецирующей.=

В зависимости от направления проецирования по отношению к плоскости проекций П' различают два вида параллельных проекций: косоугольную, когда проецирующие лучи не перпендикулярны к плоскости П', и прямоугольную или ортогональную, когда проецирующие лучи перпендикулярны к плоскости проекций.

Несмотря на то, что параллельное проецирование по сравнению с центральным дает меньшую наглядность, параллельные проекции и особенно ортогональные обладают удобоизмеримостью и простотой построен= ия. Поэтому ортогональное проецирование широко распространено в технике и являе= тся основным методом курса начертательной геометрии.

2. Основные свойства проецирован= ия

При изображении прост= ранственной фигуры на плоскость некоторые размеры и формы получаются искаженными по отношению к оригиналу. В практике же приходится по данному искаженному изображению объекта устанавливать его натуральный вид и размеры. Для этого необходимо знать законы, по которым производится проецирование, уметь перей= ти от изображения данного объекта к самому оригиналу и наоборот. Следует знать, какие элементы при данном проецировании остаются неизменными.

Свойства фигур, не изменяющиеся при проецировании, называют неизменными или инвариантными <= /i>свойствами данного проецирования. Рассмотрим основные свойства центрального и параллельного проецирования.

Общие свойства для центрального и параллельного проецирования

1. Любая точка (кроме точки 5) проектирует= ся на данную плоскость проекций в единственную точку. Если проектируемая точк= а (F) (черт. 1) принадлежит проектирующему лучу, параллельному плоскости проекций, то такой луч пересечет данную плоскость проекций в бесконечно удаленной точке F'= &y= en; , называемой несобственной точко= й.

2. Каждой точке (А, В, С, D, ...), принадлежащей какой-либо лин= ии (кривой или прямой), соотве= тствует проекция (A', В', С', D= ') этой точки на проекции данной линии (чертежи 1, 3).

3. Кривая в общем = случае проецируется в кривую, а прямая в прямую. Если прямая совпадает с проектирующим лучом, например, DЕ (черт. 1), МN (черт. 3), то она проецируется= в точку D'&o= rdm; Е''º N'. Кривая, все точки которой принадлежат проецирующей плоскости, проецируется в прям= ую.

4. Точка пересечен= ия линий проецируется в точку пересечения проекций этих линий (черт. 4), т. е., если М =3D k ´  l<= /span> или N =3D a &a= cute;  b<= /span> то М' =3D k = ;&a= cute;  l<= /span>' и N' =3D а'&= nbsp;&a= cute;  b<= /span>'.

5. Прямая, касательная к кривой линии, проецируется в касатель= ную к проекции данной кривой (черт. 5). Это свойство вытекает из свойства 4= .

Свойства, присущие= только параллельному (в том числе и ортогональному) проецированию.

6. Проекции параллельных прям= ых параллельны между собой, т. е., если а&nb= sp;|| B, то а' || = B'. Пусть отрезки АВ и параллельны (черт. 6),тогда проецирующие плоскости АА'В'В и <= span class=3DSpellE>DD'Е'Е будут также параллельны, а следовательно, линии А'В' и D'Е' пересечения этих плоскостей с П' будут параллельны.=

7. Отношение отрез= ков, принадлежащих параллельным прямым или одной прямой, равно отношению проекций этих отрезков, т. е. если АВ || , то АВ : DЕ =3D А'В': D'Е'. Возьмем д= ва параллельных отрезка АВ и DЕ (ч= ерт. 6). Через D и Е проведем взаимно параллельные прямые СD и = ВЕ, в результате ВСDЕ и В'С'D'Е' окажутся параллелограммами. Так как проецирующие лучи взаимно параллель= ны, то отношение AB\CB=3DA'B'\C'B' верно. Подставив= в это отношение вместо СВ<= /span> и С'В' равные им отрезки и D' Е , получим AB\DE=3DA'B'\D'E', следовательно,= точка, взятая на отрезке прямой, делит проекции этого отрезка в том же отношении, = что и сам отрезок, например середина отрезка проецируется в середину его проекц= ии, точка пересечения медиан проецируется в точку пересечения их проекций и т. = п.

8. При параллельном перемещен= ии плоскости проекций проекция фигуры не изменяется. Если П' || П&q= uot;, то А'В'С' =3D А"В"С&= quot; (черт. 7).

Свойства, относящи= еся только к ортогональному проецированию.

9. Отрезок прямой в общем случае равен гипотенузе прямоугольного треугольника, у которого один катет равен его проекции на данную плоскость проекций, а второй – разности расстояний концов отрезка от этой плоскости (черт. 8). Проекция А<= i>' отрезка на данную плоскость проекции равна длине этого отрезка, умноженного на коси= нус угла наклона данного отрезка к плоскости проекции, т. е. А'В' =3D АВ соsj<= /span> .

10. Любой отрезок прямой (= или кривой), параллельный плоскости проекции, проецируется на эту плоскость = без искажения, т. е., если АВ &E= acute;  т и т || П', то А<= /i>' =3D АВ, так как в данном случае соsj  =3D 1 (черт. = 9).

11. Любая плоская фигура, параллельная плоскости проекций, проецируется на эту плоскость без искажения, на= пример, если АВС || П', то А<= /i>'В'С=3D D<= /span> АВС (черт. 10).

12. Проекция любой фигуры (плоск= ой фигуры, отрезка линии и т. д.) не может быть больше самой фигуры (как следствие пп. 9 и 10).

13. Если две прямые взаимно перпендикулярны и одна из них параллельна плоскости проекций, то проекции этих прямых на да= нную плоскость проекций будут также взаимно перпендикулярны, т. е., если а&n= bsp;^<= /span>  b<= /span> и а || П', т= о а' ^<= /span>  b' (черт. 11). Пусть дано а ^  b, т.е. &E= TH;  BA= D =3D 90°<= /span> . Построим проекцию а&= nbsp;^<= /span>  b на П' и докажем,= что а' ^<= /span>  b'. AA^  П= ' (как проецирующий луч), следовательно, плоскость Г(АА'&a= cute; b= ) также перпендикулярна П'. Прямая а перпендикул= ярна плоскости Г, так как она перпендикулярна двум прямым <= i>АА' и b, принадлежащим плоскости Г. Но <= i>а' || а (а || П') и, следовательно, а' ^  Г, откуда а' перпендикулярна любой прямой плоскости Г, в том числе и b'. Отсюда справедливо, что а'&n= bsp;^<= /span>  b<= /span>', а это и требовалось доказать.

Это доказательство относится как к пересекающимся прямым, так и к скрещивающимся. Как видно из чертежа, если с &I= grave;  Г, а Г ^<= /span>  Q<= /span> ,<= /span> то с' ^<= /span>  а'.

3. Способы восполнения изображений при параллельном
проецировании

Для того чтобы по дан= ному проекционному чертежу можно было представить натуральный вид предмета и его положение в пространстве, нужно, чтобы этот чертеж был обратимым, т. е., чт= обы по нему можно было воспроизвести (изготовить) предмет.

Из рассмотренных центральных и параллельных проекций видно, что одна проекция точки, например, проекция А' точки А еще не определяет положение этой т= очки в пространстве, так как в А' может спроециров= аться любая точка, лежащая на луче ЗА (черт. 1), или на луче АА' (черт. 12). Несмотря на то, = что кривая k (черт. 13) проектируется в единственную ее проекцию k', эта проекция не определяет положение кривой в пространстве, так как = k' может соответствовать множество кривых пространства.

Чтобы проекционный че= ртеж стал обратимым, требуется дополнить его некоторыми элементами, позволяющими получить такой чертеж, по которому можно было бы определить как форму, так и размеры изображаемого предмета.

Рассмотрим некоторые основные способы восполнения ортогональных проекций.

  • Способ проекций с числовы= ми отметками

Проекции с числовыми отметками представляют собой ортогональные проекции на горизонтальную плоск= ость проекций (плоскость нулевого уровня) с указанием расстояния (числовой отмет= ки) точек от плоскости проекций.

Составленный таким об= разом чертеж вполне определяет предмет в пространстве.

 

Пусть дана горизонтальная плоскость проекций П' (плоскость нулевого у= ровня) и прямая АВ, у которой точка А расположена выше плоскости проекций (над плоскостью = П') на четыре единицы, а точка В расположена ниже плоскости проекций на 1,5 единицы (черт. 14). если ортогонально спроецировать точки А <= /i>и В на плоскость П' и проставить их числ= овые отметки (высоты) относительно плоскости П' с указанием положительных (выше П<= /span>') и отрицательных (ниже П') значений, то получим изображение, вполне определяющее как форму предмета, так и его размеры (черт. 15).

Способ проекций с чис= ловыми отметками применяется при изображении пространственных форм, у которых одно измерение (в вертикальном направлении) очень мало по сравнению с большой протяженностью по двум другим направлениям (в горизонтальном направлении). К таким поверхностям следует отнести земную поверхность. Земную поверхность изображают семейством горизонталей с указанием их числовых отметок. За плоскость проекций принимается уровень моря.

Проекции с числовыми отметками применяют и при изображении других сложных пространственных форм,= в том числе и в химии.

  • Способ аксонометрических проекций

   =     Аксонометрическая     проекция – один из способов изображе= ния пространственных форм на плоскости. Этот вид проекций обладает большой наглядностью и является обратимым изображением (черт. 16).

Сущность способа аксонометрических проекций состоит в том, что изображаемый объект (А), <= /i>отнесенный к системе координат трех взаимно перпендикулярных осей Oxyz (натуральной системы координат) вместе с этой системой проектируется на одну плоскость проекции П' (картину). Для удобства измерения по всем трем осям откладывают отрезки, ра= вные отрезку е, который называют масштабным отрезком или натура= льным масштабом. Полученные проекции А' и А'1 точки А вместе с осями О'х'у'z' и масштабными отрезками е'х = е'у , е'z = определяют положение точки А в пространстве. = А' – называют аксонометрической проекцией точки A; A'1вторичной проекцией точ= ки А или аксонометрической проекцией горизонтальной проекции А1= точки А.

Аксонометрические про= екции используют при построении наглядных изображений.

  • Способ проекций на компле= ксном чертеже

Как известно, одна параллельная проекция не определяет проецируемый предмет в пространстве. Для того чтобы предмет был вполне определен в пространстве, достаточно построить две его проекции.

Отнесем некоторую точ= ку А к двум взаимно перпендикулярным координатным плоскостям П1 и = П2 (черт. 17), расположив П1 горизонтально, а П2фронтально перед наблюдателем. Плоскость П1 назовем горизонтальной, а плоскость П2 – фронтальной плоскостью проекций. Ортогональные проекции точки А на эти плоскости проекций назовем соответственно горизонтальной А1 и фронтальной А2 проекциями.

Две ортогональные про= екции А1 и А<= i>2 на данные плоскости проекций определяют положение точки А в пространстве, так как отрезок А1Ах определяет расстояние от точки А до плоскости П2 (глубину), а отрезок А2АХ – расстояние от точки А до плоскости П= 1 (высоту).

Для более удобного построения ортогональных проекции объекта строят плоский чертеж двух проекц= ий. Такой чертеж можно получить путем вращения одной из плоскостей проекций вок= руг линии их пересечения * до совмещения со второй плоскостью проекций. Наприме= р, вращением плоскости проекций П1 вокруг линии х можно совместить е= е с плоскостью проекций П2 (черт. 1= 8).

После совмещения плос= костей (П1 = &o= rdm;  П<= sub>2) проекции А1 и <= i>А2 будут лежать в одной плоскости чертежа на линии связи, перпендикулярной к оси проекций. Так как линия пересечения плоскостей х является одновременно проекцией на две плоскости, то ее обе проекции совпадут. Плоский чертеж, состоящий из двух (= или более) взаимосвязанных проекций А1 и А2 , = называют комплексным чертежом точки А или эпюрой Монжа.

Комплексные чертежи ш= ироко используют в технике.

Виды изображений. = Смотря на какой-либо предмет, = мы можем составить суждение о его форме, размерах, = положении в пространстве, взаиморасположении отдельных частей и т. д. Можно прийти к суждению о предмете и по его изображению, однако только в том случае, когда изображение обладает определенными качествами. К ним относятся:<= /span>

  • обратимость – свойство изображения, позволяющее по нему однозначно восстановить действительную форму и размеры предмета, а также его положение в пространстве; это свойство служит тому, чтобы по изображению можно было изготовить предм= ет, например по проекту здания возвести его в натуре; графическое изображение, обладающее свойством обратимости, называется чертежом; обратимость связана с той информацией, которую может обеспечить чертеж; информация может быть графической, аналитической и словесной; начертательная геометрия в пер= вую очередь использует графическую информацию;
  • наглядность – свойство изображения, дающее возможность вызвать по нему в мозгу зрителя пространственное представл= ение о предмете; важным элементом наглядности является естественность; <= /i>естественное изображение создает зрительное суждение, близкое к тому, которое возни= кает при рассматривании самого предмета; естественность особенно важна в архитектурно-строительной практике, когда по изображению оцениваются эстетические достоинства проектируемого объекта;
  • единство условностей, принятых при выполнении изображения; они должны быть такими, чтобы каждый мог “прочесть” изображение, выполненное другим лицом.

Самым естественным изображением является перспектива (рис. 1). На перспективном изображении наблюдаются те явления, с которыми мы встречаемся в натуре: параллельные линии кажутся пересекающимися, предметы по мере их удаления от зрителя – уменьшающимися и т. п. Такое изображение, однако, обладает серьез= ным недостатком: без дополнительных, иногда довольно сложных, построений по нему нельзя установить размеры изображенного предмета.

Изображения, выполнен= ные с учетом некоторых условностей, даны на рис. 2 и 3. На рис. 2 показана акс= онометрия, на рис. 3 – ортогональные проекции (три определенным образом связанные между собой изображения).

Перспектива используе= тся в основном в архитектурно-строительной практике и дорожном строительстве, аксонометрия и ортогональные проекции – во всех областях науки и техники. Аксонометрия обладает большей наглядностью, чем изображения, выполненные в ортогональных проекциях, но меньшей, чем перспектива. По аксонометрическому изображению можно проводить некоторые измерения непосредственно на чертеже. Ортогональные проекции наименее наглядны, но очень удобны для решения ме= трических, (т. е. связанных с измерениями) задач.

При изложении дальней= шего материала мы будем часто прибегать к аксонометрии, рассматривая ее как сред= ство выполнения наглядного, удобного в построении изображения. Теория же аксонометрических изображений будет рассмотрена несколько позднее.


= 1. Практикум по инженерной графике<= o:p>

1.1. Указан= ия к практикуму

Методические указания= по инженерной графике содержат:

= 1.&n= bsp;    мини-графы к изучаемым темам основных разделов лекционного курса;

= 2.&n= bsp;    тесты для 1 и 2 уровней усвоения для подготовки к практическим занятиям и эталоны (ответы);

= 3.&n= bsp;    условия для решения задач, необходимых для выполнения индивидуальных расчетно-графических работ.

Мини-граф по соответствующей теме (15–20 учебных элементов) студенту рекомендуется заучи= ть наизусть, что дает возможность ориентироваться в основных вопросах темы.

На каждом практическом занятии отводится 5 мин. для воспроизведения соответствующего мини-графа по памяти. Еще 5 мин. тратится на взаимопроверку или на проверку качества воспроизведения мини-графов преподавателем. По возможности применяются компьютерные технологии.

Вся остальная часть практического занятия посвящена изучению учебных элементов, содержащихся в мини-графе, в процессе решения задач и выполнения расчетно-графических рабо= т на 3 уровне усвоения (знаний, умений, убеждений).

При подготовке к прак= тическому занятию студент имеет возможность проверить качество своих знаний на 1 и 2 уровнях усвоения (узнавания и памяти) с помощью соответствующих тестов.

Тест (проба, испытани= е) представляет собой сочетание задания и правильного его решения (эталона).

Степень готовности к практическому занятию студент определяет, сопоставляя свой ответ с эталоном= .

Помещенные в методиче= ские указания задачи должны быть решены каждым студентом на практических занятиях или самостоятельно.

При защите очередного задания, предусмотренного рабочей программой курса инженерной графики, студ= ент должен представить свою тетрадь с решенными задачами, уметь обосновать выбранный способ их решения и пояснить ход решения.

На последнем практиче= ском занятии студент получает допуск к экзамену, если выполнены и сданы расчетно-графические работы, а также тетрадь с конспектом лекций и решенными задачами.

Задачи решают согласно текстовым условиям и графическому заданию.

Все построения должны= быть выполнены аккуратно при помощи чертежных инструментов и принадлежностей или автоматизированным способом на ЭВМ.

Линии связи и другие = линии построения проводят тонкими линиями с нанесением на них стрелок, поясняющих= ход решений. Полученные результаты решения (точки, линии, фигуры и т.д.) обводят линиями основного контура.

Во всех случаях, когд= а это возможно, задачи решать на безосных чертежах. Все линии построений, использовавшиеся при решении задач, необходимо сохранить.=

При решении задач необходимо составлять алгоритмы их решения.

Раздел 1.

Тема 1. Изо= бражение точки на эпюре монжа

1.1. Построить нагляд= ное изображение точек А,С,D,Е= , заданных координатами, обозначить две проекции каждой точки в системе. Построить комплексный чертеж (эпюр) точек по образцу точки А (38,20,25).

Построить наглядное изображение и эпюр точки К, удаленной от плоскос= тей проекций П1,П2 и Пз, соответственно на 10 мм, 30 мм и 70 мм.

Определить положение = точки N, расположенной над точкой К на расстоянии 20 мм и точки М, расположенной за точкой К на расстоянии 10 мм. Отметить видимость точек на плоскостях проекций П1 и П2 .=

1.2. Построить проекц= ии точек А, В, С на плоскость = Пз проекционным способом.

 

Тесты для 1 и 2 уровней усвоения

Задача 1.1.

1. = Каждая из данных точек А, В, С, D<= /span>, Е однозначно определена в пространстве?

2. = Какие отрезки на чертеже соответствуют координатам Х,У,Z точки А?

3. = Какие точки принадлежат горизонтальной плоскости проекций?

4. = Какие точки принадлежат фронтальной плоскости проекций?

5. = Какие точки принадлежат осям проекций?

6. Определено ли на чертеже положение профильной плоскости проекций?

7. = Как образован эпюр Монжа точки А? =

8. = Запишите координаты точек К,N,М.

9. = Какая из точек равноудалена от плоскостей проекций П1 и П2

Задача 1.2.

1. Определяет ли точка 0 положение профильной плоскости проекций Пз?

2. Определите по чертежу, какие точки принадлежат плоскостям проекций.

3. = Какими осями можно обозначить на чертеже плоскость Пз?=

 

Раздел 1.

Тема 2. Лин= ии: прямая и кривая. Две прямые задачи

2.1. Достроить горизонтальную и профильную проекции пирамиды 5 АВС так, чтобы ребро АС было параллельно плоскости Пг , а ребро АВ расположено параллельно П1 и под углом b =3D 30 к П2.

Определить видимость = ребер на проекциях, если считать грани пирамиды непрозрачными.<= /p>

2.2. Отрезок АС (А2С2; А1С1)|| П2 – диагональ ромба АВСD. Вершина В принадлежит плоскости П1, вершина D равноудалена от плоскости П1 и П2.

Построить проекции ро= мба.

2.3. На линии = t от точки А отложить о= трезок АВ =3D 50 мм. Определить проекции этого отрезка= .

 

Тесты для 1 и 2 уровней усвоения

Задача 2.1.

1. = Какое ребро пирамиды является горизонталью?

2. = Какое ребро пирамиды является фронталью? <= /span>

3. = Какое положение в пространстве занимает ребро АS?

4. = Какие ребра пирамиды занимают скрещивающееся положение?

5. Определено ли положение постоянной чертежа Ко? <= o:p>

Задача 2.2.

1. = Какая фигура называется ромбом?

2. = Как расположены диагонали ромба относительно друг друга?

3. = Можно ли найти точку пересечения диагонали ромба?

4. = На какую из плоскостей проекций проецируется угол между диагоналями ромба в натураль= ную величину?

5. = Какой линии принадлежит фронтальная проекция точки В? =

6. = Какие координаты точки D будут одинаковыми?

Задача 2.3.

1. = Какое положение в пространстве занимает линия t= ?

2. = Можно ли на какой-либо из данных проекций линии t отложить отрезок АВ = =3D 50 мм?

3. = Сколько решений имеет задача?

4. = Можно ли на прямой t отметить отрезок любой длины и определить его натуральную величину?

5. = Как определить разность высот двух точек отрезка?

6. = Как определить разность глубин двух точек отрезка?

Задача 2.1

 

Задача 2.2

Задача 2.3

Раздел 1.

Тема 3. Плоскость. Линии и точки в плоскости.
Перпендикуляр к плоскости

Задача 3.1.

 

     3.1. Шар 0 (01,02) кати= тся по наклонному щиту АВСD. Определить положение шара 0'(0'1,02) в момент, когд= а он коснется поверхности земли, а также угол наклона щита к земле.

     3.2. Построить проекции пирамиды SАВС (S2А2В2С2 ,S1а1B1С1) если высота ее 50 =3D 60 мм, а точка 0 (02,01) – центр описанной окружнос= ти около основания АВС. Показать видимость ребер= .

    3.3. Построить пр= оекции пирамиды SАDС, есл= и высота ее =3D 60 мм, а основание – равнобедренный треугольник АВС, принадлежащий плоскости Е (<= span class=3DSpellE>hÇ f). Вершина АÌ h, вершина ВÌ f, АС =3D ВС =3D 50 мм. Показать один из возможных вариантов. Определить видимость реб= ер пирамиды.

Тесты для 1 и 2 уровней усвоен= ия

Задача 3.2.

 

    1. Какое положение в пространстве занимает основание пирамиды АВС?

   2. Какое положение занимает высота пира= миды S0, если пирамида прямая ?

   3. Ка= кая точка является центром описанной окружности треугольника?

 

Задача 3.3.

 

  1. На каких линиях на чертеже можно отложить натуральную величину отрезков СА =3D СВ =3D S0 мм ?

  2. Как направлено ребро = SС пирамиды ?

  3. Можно ли отложить высоту пирамиды SС =3D 60 мм на какой-ли= бо ее проекции?

  4. Как на п= рямой (высоте пирамиды) от точки С отложить отрезок = СS =3D 60 мм.

 

Рис. 4. Образование и изображение поверхностей на эпюре монтажа: а – исходные данные; б – наглядное изображение; в – ортогональные п= роекции (эпюр Монжа)

 

Раздел 1.

Тема 3. Пов= ерхности, их виды, задание и изображение на чертеже.
Точки и линии на поверхности

3.4, 3.5. Построить проекции поверхности Е (1,), заданной определителем. Записать название поверхности.

3.6. Построить проекц= ии поверхности. Обозначить проекции очерковых образующих на П1 и П2. Показать видимость. Построить горизонтальную проекцию точки А (А2) , принадлежащей поверхности.

Записать название поверхности.

3.7. Построить проекц= ии отсека поверхности, заданной определителем. Построить недостающую проекцию линии а, принадлежащей данной поверхности. Записать название поверхности.

 

Тесты для 1 и 2 уровней усвоения

Задачи 3.4, 3.5.

1. = Какие элементы записаны в геометрической части определителя?

2. = Каково взаимное расположение оси l и линии?

3. = Как прочитать алгоритмическую часть определителя?

4. = По чертежу и определителю установить название поверхностей. =

Задача 3.6.

1. = Какими геометрическими элементами определена поверхность?

= 2. Как прочитать алгоритмическую = часть определителя поверхности?

3. = Можно ли построить очерк поверхности?

4. = По чертежу и определителю установить название поверхности. <= /p>

5. = Когда точка принадлежит поверхности?

Задача 3.7.

1. = Как прочитать геометрическую часть определителя?

2. = Как прочитать алгоритмическую часть определителя?

3. = По чертежу и определителю установить название поверхности. <= /p>

4. = Очерком или отсеком изображена на чертеже поверхность?

 

 

 

Раздел 1.

Тема 4. Спо= собы преобразования комплексного чертежа.
Метрические задачи

4.1. Определить положение центра окружности, вписанной в D<= /span> АВС способом замены плоскостей проекций.

4.2. Определить расстояние АК ( А2К21К) от точки А (А2,А= ) до плоскости å (D СDЕ).

4.3= . Под каким углом к горизонтальной плоскости течет вода в лотке водостока, установленном на плоском откосе по направлению S. Определить длину лотка ме= жду точками А и В.

4.4. Способом плоскопараллельного движения и способом замены плоскостей проекций определить величину двугранного угла при ребре АВ.

4.5. Способом замены плоскостей проекций определить расстояние между двумя параллельными прямыми а|| в.

4.6. Способом замены плоскостей проекций определить расстояние между скрещивающимися прямыми m-п.

Построить проекции об= щего перпендикуляра в системе П2= \П1.

Тесты для 1 и 2 уровней усвоения

Задача 4.1.

1. = Какая точка является центром вписанной окружности в треугольнике?

2. = Можно ли на каком-либо изображении треугольника АВС пост= роить биссектрисы углов?

3. = Какие из основных задач преобразования чертежа имеют место?

Задача 4.2.

1. = Каким отрезком определяется расстояние от точки А до плоскости СDЕ?

2. = Можно ли без дополнительных построений провести проекции перпендикуляра? =

3.Какая из основных задач преобразования чертежа имеет место?

Задача 4.3.

1. = Какое положение в пространстве занимает линия откоса АВ?

2. = Между какими линиями заключен угол наклона прямой к плоскости? =

3. = Какая из основных задач преобразования чертежа имеет место?

Задача 4.4.

1. = Какой угол называется двугранным?

2. = Каким углом измеряется двугранный угол?

3. = Какие из основных задач преобразования чертежа имеют место?

Задачи 4.5, 4.6.

1. = Какой отрезок является кратчайшим расстоянием между двумя прямыми?

2. = Сколько решений имеет задача?

3. = Какие из основных задач преобразования чертежа имеют место?

 

 

 

 

 

Раздел 1.

Тема 5. Поз= иционные задачи пересечение плоскости и поверхности

5.1, 5.2. Построить проекции линии пересечения поверхности и плоскости. Показать видимость. Обозначить опорные точки. Определить вид фигуры, лежащей в секущей плоскост= и.

5.3. Построить след проецирующей плоскости, которая пересекает заданную поверхность по гипербол= е. Построить проекции и натуральную величину фигуры сечения. Обозначить опорные точки, указать видимость линии сечения.

5.4. Построить проекц= ии и натуральную величину сечения конуса плоскостью å , проходящей через точку А (= А21) и пересекающей конус по параболе.

Пересечение поверхнос= тей Способ вспомогательных секущих плоскостей. Способ сфер.

5.5, 5.6. Построить проекции линии пересечения поверхностей, обозначить опорные точки. Показать видимость линии пересечения и очерков данных поверхностей.

5.7. Построить проекц= ии линии пересечения поверхностей, образующих корпус фитинга (тройника), служа= щего для соединения труб. Обозначить опорные точки. Показать видимость.

Изображение фитинга выполнено условно.

5.8. Построить проекц= ии линии пересечения поверхностей, ограничивающих корпус крана, используя част= ный случай пересечения поверхностей второго порядка. Обозначить опорные точки. Показать видимость.

Изображение корпуса к= рана выполнено условно.

Тесты для 1 и 2 уровней усвоения

Задачи 5.1, 5.2.

1. = Какие поверхности пересекают плоскость å ?

2. = Как расположена в пространстве секущая плоскость å ?<= /span>

3. = Какого вида фигура получается при сечении данных поверхностей плоскостью?

4. = Какой вид имеют проекции сечения на П1 и П2?

5. = Какие виды опорных точек требуется определить?

Задачи 5.3, 5.4.

1. Перечислить кривые конических сечений.

2. = Сколько решений имеют задачи?

Задачи 5.5, 5.6, 5= .7, 5.8.

1. = Какие поверхности пересекаются?

2. = Известна ли хотя бы одна проекция линии пересечения поверхностей? =

3. = Каким способом решается задача?

 

РАЗДЕЛ 1.

ТЕМА 6. Позиционные задачи.
Пересечение прямой с плоскостью и поверхностью

5.9= –5.11. Построить проекции точек пересечения прямой с данной поверхностью.

Пок= азать видимость прямой относительно поверхности.

5.1= 2. На какой глубине при вертикальном бурении из точки D поверхности земли встретится пласт, плоскость которого определяется точками= А, В, С.

5.1= 3. Определить точки, в которых мачта АВ антенны и = ее растяжки АС, АD и АЕ пересекают кровлю.

Показать видимость мачты и растяжек.

 

Тесты для 1 и 2 уровней усвоен= ия

Задачи 5.9–5.11.

1. = Какое положение занимает линия l в пространств= е?

2. = Сколько общих точек имеют прямая l и данная поверхность?

3. Обозначить имеющиеся проекции точек пересечения в данных задачах.

Задача 5.12.

1. Обозначить на чертеже проекции линии бурения l(= l2,l1).

2. = Имеет ли место частный случай решения?

Задача 5.13.

1. = Какое положение в пространстве занимают антенна и растяжки?

2. = Сколько скатов кровли занимает частное положение относительно = П1 и П2?

3. = Для какого элемента мачты имеется решение на чертеже?

 

 

 

 

2. ЭТАЛОНЫ

Ответы на тесты 1 и 2 уровней усвоения

= Тема 1

Задача 1.1.

1. Да.

2. ХА =3D = ОАх; Yа =3D A1Aх; ZА =3D АхА2.

3. В, D, Е.

4. С, Е.

5. D Ì<= span style=3D'font-size:13.5pt;font-family:Arial'> OУ; Е Ì ОХ.

6. Да.

7. Совмещенное положе= ние плоскостей проекций с изображением на них точки А.

8. К (70; 30; 10); N (70; 30; 30); М (70; 20; 10).

9. N

Задача 1.2.=

1. Да.

2. В, С.

3. ОZ Ç = ОУ.

Тема 2=

Задача 2.1.

1. АВ || П1

2. АС || П2

3. Общее положение.

4. АС – SВ; ВСАS; АВ.

5. Да, через точку S0.

Задача 2.2.

1. Ромб – это параллелограмм, у которого все стороны равны, а углы непрямые.

2. Перпендикулярно.

3. Да (А2O2 =3D O2С2 или А1O1 =3D O1С1)

4. На П2.

5. Оси ОХ.=

6. YD=3DZD

Задача 2.3.

1. l – общего положения.

2. Нет.

3. Два= .

4. Да<= span lang=3DPL style=3D'font-size:13.5pt;font-family:Arial;mso-ansi-language:PL'= >.

5. D<= span lang=3DPL style=3D'font-size:13.5pt;font-family:Arial;mso-ansi-language:PL'= > Z =3D Z1 – Z2.

6. D<= span lang=3DPL style=3D'font-size:13.5pt;font-family:Arial;mso-ansi-language:PL'= > Y =3D Y1 – Y2.

Тема 3=

Задача 3.2

1. D АВС||П<= sub>1; D АВС^ П2

2. S0^ D АВС^ П1; SО||П2<= /sub>

3. Точка пересечения срединных перпендикуляров сторон треугольника.

Задача 3.3.

1. (С2В2 Ì<= span style=3D'font-size:13.5pt;font-family:Arial'> f2<= /sub>) =3D 50 мм; ( С1А1 Ì С h1) =3D 50 мм

2. SС АВС

3. Нет, так как SС – общего положения.

4. См. решение задачи= 2.3.

Тема 3=

Задачи 3.4, 3.5.

1. Ось i и линия l.

2. Задача 3.4. i Ç l= .

3. 1) ljнекоторое положение линии l;

2) Ri – вращение вокруг оси i;

3) li-данная образующая, т.е. некоторое положение линии I (lj) определяется вращением вокр= уг оси i (Ri) д= анной образующей l.

4. Задача З.4. – конус вращения;

Задача З.5. – поверхность вращения (конус, тор)

Задача 3.6.

1. Кривая m и прямая s (направление= ).

2. Любая образующая пересекает линию m и параллельна прямой s.

3. Да.

4. Цилиндрическая поверхность.

5. Точка принадлежит поверхности, если она расположена на линии, принадлежащей этой поверхности.=

Задача 3.7.

1. Поверхность, задана линиями m, n и плос= костью параллелизма П1<= /span>.

2. Образующая l движется одновременно по двум направляющим л= иниям тип параллельно плоскости П1<= /sub>.

3. Коноид.=

4. Отсек.<= /span>

 

 

Тема 4=

Задача 4.1.

1. Точка пересечения = биссектрис углов.

2. Нет.

3. Третья и четвертая основные задачи.

Задача 4.2.

1. Перпендикуляром от= точки А до пересечения его с треугольником.

2. Нет.

3. Третья основная за= дача.

Задача 4.3.

1. АВ – общего положения.

2. Угол заключен между самой прямой и ее ортогональной проекцией на эту плоскость.

3. Первая основная за= дача.

Задача 4.4.

1. Угол между двумя плоскостями.

2. Линейным углом, ле= жащим в плоскости, перпендикулярной к ребру.

3. Первая и вторая ос= новные задачи.

Тема 5=

Задачи 5.1, 5.2.

1. Пирамида (5.1); сф= ера (5.2);

2. å ^ П2 (= 5.1); å ^ П1 (5.2);

3. Четырехугольник (5.1);окружность (5.2);

4. На П2отрезок; на П1 – четырехугольник (5.1); на П1 – отрезок; на П2 – эллипс (5.2);

5. Точки перелома лин= ии, принадлежащие ребрам пирамиды (5.1); Точки – границы видимости, принадлежащ= ие очерковым линиям; высшая и низшая точки, принадлежащие общей плоскости симметрии (5.2).

Задачи 5.3 ,5.4.

1. Окружность, эллипс, гипербола, парабола.

2. Множество (5.3); о= дно (5.4).

Задачи 5.5, 5.6, 5= .7, 5.8.

1. Усеченный конус, п= рямая призма (5.5). Конус вращения, цилиндр вращения (5.6). Два цилиндра вращения (5.7). Усеченный конус, цилиндр вращения (5.8).

2. Да, наП2 (5.5); да, на П2 (5.6); да, на П1 (5.7); н= ет (5.8).

3. Способ секущих плоскостей (5.5, 5.6);

Способ секущих плоско= стей и способ сфер (5.7, 5.8).

Тема 6=

Задачи 5.9–5.11. <= o:p>

1. l^ П1= (5.9): l- общего положения (5.10), l || П1(5.11).

2. Две точки.

= 3.Точка, принадлежащая основанию конуса(5.9)

Задача 5.12.<= /o:p>

1. l2 É<= span style=3D'font-size:13.5pt;font-family:Arial'> D2<= /sub>D1; l1=3DD1.

2. Да, так как l ^ П1.

Задача 5.13.<= /o:p>

1. Антенна АВ П1; растяжка АС||П2; растяжки АD.АЕ – общего положения.

2. Один скат.

3. Для растяжки АС.

Список литературы

= 1.&n= bsp;    Романычева, Э.Т. Инженерная и компьютерная графика / Э.Т. Романычева, Т.Ю. Соколов= а, Г.Ф. Шандурина. – М.: ДМК<= /span> Пресс, 2001.

= 2.&n= bsp;    Чекмарев, А.А. Инженерная графика / А.А. Че= кмарев. – М.: Высш. шк., 19= 98.

= 3.&n= bsp;    Начертательная геометрия. Инженерная и машинная графика / Под р= ед. К.П. Валькова. – М.: Высш<= /span>. шк., 1997.

Тамара Станиславовна Красовская

ИНЖЕНЕРНАЯ ГРАФИКА

Методические указания изучения дисциплины
и учебно-методический материал для самостоятельной работы
=

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План 2002 г. Поз. 3.36.

Технический редактор И.А. Нильмаер.

Отпечатано с авторских оригиналов.

ИД № 05247 от 2.07.2001 г. ПЛД № 79-19 от 19.01.2000= г.

Подписано в печать 10.09.2002. Печать офсетная. Бумага тип. № 2.

Формат 60х841/16. Усл. печ. л. 2,8. Зак. 1= 32. Тираж 150 экз. Цена 19 р.

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Издательство ДВГУПС

680021, г. Хабаровск, ул. Сер= ышева, 47

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