Piv txwv teeb meem tham txog kev sib hloov lej
Pendahuluuan
Kev tig yog ib qho kev hloov pauv geometric uas feem ntau ntsib hauv kev suav lej, tshwj xeeb yog geometry. Kev tig yog kev tig ib yam khoom ncig ib qho chaw tshwj xeeb (qhov chaw nruab nrab ntawm kev tig) los ntawm lub kaum sab xis tshwj xeeb hauv kev taw qhia clockwise lossis counterclockwise. Lub tswv yim ntawm kev tig yog qhov tseem ceeb hauv cov teb xws li computer graphics, physics, thiab engineering. Tsab xov xwm no yuav tshawb nrhiav ntau yam piv txwv thiab kev sib tham ntawm kev tig hauv kev suav lej.
Nkag Siab Txog Kev Tig
Kev tig yog ib qho kev hloov pauv uas txav txhua qhov chaw ntawm ib yam khoom los ntawm kev tig nws ncig ib qho chaw ruaj khov, hu ua qhov chaw nruab nrab ntawm kev tig, los ntawm ib lub kaum sab xis hauv ib qho kev taw qhia. Cov cim qhia dav dav rau kev tig nrog lub kaum sab xis θ thiab qhov chaw nruab nrab ntawm kev tig (a, b) tuaj yeem sau ua R_(a, b)(θ).
Rau ib qho chaw P(x, y) tig los ntawm θ degrees nrog qhov chaw nruab nrab ntawm kev tig ntawm qhov pib (0, 0), cov kev sib koom ua ke tshiab ntawm P'(x', y') tom qab kev tig yuav tau txais los ntawm kev siv cov mis:
– x' = x cos θ – y sin θ
– y' = x sin θ + y cos θ
Cia peb txuas ntxiv nrog qee qhov piv txwv ntawm cov teeb meem kev sib hloov ua tiav nrog lawv cov kev sib tham.
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Piv txwv lus nug 1
Lus Nug: Txheeb xyuas cov kev sib koom ua ke tshiab ntawm qhov A (3, 4) tom qab tig nws 90 degrees counterclockwise nrog qhov chaw ntawm kev tig ntawm lub hauv paus chiv keeb (0, 0).
Kev Sib Tham: Siv cov qauv kev sib hloov nrog lub kaum sab xis 90 degree counterclockwise:
– x' = x cos(90°) – y sin(90°) = 3(0) – 4(1) = -4
y = x sin(90°) + y cos(90°) = 3(1) + 4(0) = 3
Yog li, cov kev sib koom ua ke tshiab ntawm A' tom qab kev sib hloov yog (-4, 3).
Piv txwv lus nug 2
Lus Nug: Point B (2, -1) tig 180 degrees hauv ib qho kev taw qhia clockwise nrog lub hauv paus ntawm kev tig tseem nyob ntawm qhov pib (0, 0). Txheeb xyuas cov kev sib koom ua ke tshiab ntawm point B tom qab kev tig.
Kev Sib Tham: Kev tig 180 degrees mus rau sab xis lossis sab laug muab cov txiaj ntsig zoo ib yam, uas yog cov kev sib koom ua ke ntawm qhov taw tes hloov mus rau (-x, -y).
– x' = -x = -2
– y' = -y = 1
Yog li, cov coordinates tshiab ntawm B' yog (-2, 1).
Piv txwv lus nug 3
Lus Nug: Ib qho C(-3, 5) tig 270 degrees counterclockwise nrog qhov chaw nruab nrab ntawm kev tig ntawm lub hauv paus chiv keeb (0, 0). Txheeb xyuas qhov C tom qab kev tig.
Kev Sib Tham: Kev tig 270 degree counterclockwise yog sib npaug rau kev tig 90 degree clockwise.
- x' = x cos(90°) + y sin(90°) = -3(0) + 5(1) = 5
- y' = -x sin(90°) + y cos(90°) = -(-3)(1) + 5(0) = 3
Yog li, cov kev sib koom ua ke tshiab ntawm C' tom qab kev sib hloov yog (5, -3).
Piv txwv lus nug 4
Lus Nug: Txheeb xyuas cov kev sib koom ua ke tshiab ntawm qhov taw tes D (5, 5) tom qab tig nws 45 degrees nrog qhov chaw nruab nrab ntawm qhov pib (0, 0).
Kev Sib Tham: Siv cov qauv kev sib hloov nrog lub kaum sab xis 45 degree:
x' = x cos(45°) – y sin(45°) = 5(cos(45°)) – 5(sin(45°)) = 5(√2/2) – 5(√2/2) = 0
y = x sin(45°) + y cos(45°) = 5(√2/2) + 5(√2/2) = 5√2/2 + 5√2/2 = 5√2
Yog li, cov kev sib koom ua ke tshiab ntawm D' tom qab kev sib hloov yog (0, 5√2).
Kev Tig Nrog Chaw Tig Tsis Yog Ntawm Keeb Kwm
Kev tig tsis yog ib txwm ua txog qhov pib. Piv txwv li, xav tias peb xav tig ib qho chaw nrog qhov chaw nruab nrab ntawm kev tig (h, k). Yuav ua li no, peb yuav tsum kho cov kev sib koom ua ke raws li nram no:
1. Txhais lub ntsiab lus kom (h, k) dhau los ua keeb kwm.
2. Siv cov qauv kev sib hloov.
3. Txhais rov qab mus rau qhov chaw qub.
Piv txwv lus nug 5
Lus Nug: Point E(5, 7) tig 90 degrees counterclockwise nrog lub plawv ntawm kev tig ntawm point (2, 3). Txheeb xyuas cov coordinates tshiab ntawm point E tom qab kev tig.
Kev Sib Tham:
1. Txhais qhov chaw E mus rau qhov keeb kwm piv rau qhov chaw nruab nrab ntawm kev tig (2, 3):
– Lub ntsiab lus tshiab E' = (5 – 2, 7 – 3) = (3, 4)
2. Tig 90 degrees counterclockwise ncig lub taw tes tshiab:
- x' = 3 cos(90°) – 4 sin(90°) = -4
y = 3 sin(90°) + 4 cos(90°) = 3
Yog li, cov kev sib koom ua ke tom qab tig yog (-4, 3).
3. Txhais rov qab mus rau qhov chaw qub piv rau qhov chaw nruab nrab ntawm kev tig (2, 3):
– Qhov kawg E' = (-4 + 2, 3 + 3) = (-2, 6)
Yog li, cov kev sib koom ua ke tshiab ntawm qhov E tom qab kev sib hloov yog (-2, 6).
Xaus
Kev tshuaj xyuas thiab nkag siab txog kev sib hloov lej yog qhov tseem ceeb heev rau ntau yam kev siv. Los ntawm cov piv txwv thiab kev sib tham saum toj no, cov nyeem ntawv yuav tsum nkag siab txog kev ua haujlwm ntawm cov qauv sib hloov thiab lawv tuaj yeem siv rau ntau qhov xwm txheej li cas. Qhov kev tawm dag zog no tsis yog tsuas yog txhawb nqa kev suav lej yooj yim xwb tab sis kuj tseem muaj txiaj ntsig zoo rau lwm qhov chaw uas muaj kev hloov pauv geometric.
