In most pigeon yoga pose variations one leg reaches to the back while the other leg is forwards with the knee bent and the shin on the floor. Viewed from the front the front knee is generally positioned to the outside of the hip.
One option for pigeon pose is to keep the hips level from left to right. This usually means that both hips are off of the floor. The intent can be to sink the hips by reaching the back leg further back.
Another option is to do pigeon yoga pose with the hip of the front leg resting on the floor. You may have to lean to the side in order to do this. With the hip on the floor you can then position the shin of the front leg parallel to the front edge of your mat. Here the intent can be to move the back leg hip closer to the floor.
This latter variation places less stress on the front knee but can feel quite intense. Formerly I called this variation "screaming pigeon". I now tend to think of it as "Intense Pigeon Pose".
When teaching the non-intense version of pigeon yoga pose I often start with upright pigeon because it is easier to learn how to use the front leg and knee.
Position your front knee to the outside of your front hip so that your thigh angles outwards from your hip. About 4 to 6 inches is fine. (If you have your knee too far out to the side then you change the feeling of the pose.)
Position your foot about half way forwards between knee and hip. To increase the hip stretch move your front foot further forwards. Be cautious since this can place more stress on the knee particularly if you keep your front ankle extended (bent backwards). To reduce the hip stretch move your foot back. Try to pull your pelvis back from your front knee as opposed to pushing forwards into it.
The first exercise is to help you learn to activate the hip muscles of your front leg. Start with your right knee forwards and your right hip on the floor.
Then, lean your ribcage to the left to the point where your right hip comes off of the floor. Initially use your hands to help, then see if you can lift your hip up without using your hands.
Rather than leaning to the left and balancing to lift the hip, another variation is to focus on pressing the front knee down into the floor so that the muscles of the right hip and thigh activate. Your right hip should lift as a result.
Once you can comfortably lift the hip vary the knee pressure so that your right hip is level with your left hip.
Another exercise that you can do in pigeon yoga pose with your torso upright is push your pelvis back and down by pressing your front knee forwards against the floor. The feeling is like you are pushing your front knee forwards, but because it is on the floor, or on your mat, the mat stops it from slipping and so you end up pushing your pelvis back.)
Once you've got the feel for pushing your pelvis back using your front leg, then reach your back knee rearwards away from your hip. If your toes are pointing back then reach your toes back. If your toes are tucked under then reach back through your heel.
With your base sorted out you can now focus on using your arms to help push your ribcage upwards and back. So that your whole body works together, focusing on creating space. Use your front knee to push your pelvis back and down. Reach your back leg back, away from your front knee. Reach your ribcage upwards and then rearwards (bend your spine backwards) away from your pelvis.
To maximize the bend in your spine try first to tilt your pelvis forwards so that your lumbar spine bends backwards. Then try to bend your thoracic spine backwards also, opening the front of your ribcage. With your spine bent backwards, then work at tilting your pelvis backwards to open the front of your front leg hip.
If you've ever done downward dog, and I'm assuming you have, you know that you can use your shoulders to push your ribcage away from your hands. If in turn you expand your ribcage, you can push your pelvis away from your ribcage. Normally I teach people to push back while inhaling and then to relax while exhaling.
In the laying down version of pigeon yoga pose you can do the same thing.
From upright pigeon pose, reach your hands forwards to the point that your torso is about 45 degrees off of the floor. Use your hands to push your ribcage up and back and transfer that force to your pelvis. Use both your arms and your front leg to push your pelvis back as you inhale. Relax as you exhale.
Lay down on the floor fully with your arms ahead of your. Bring your hands back just enough that you can still use your hands to push your body back as you inhale. As you inhale push back with your hands and knee and as you exhale slowly relax down. Try to keep your pelvis level from left to right.
You might want to do the laying down version of pigeon pose two or three times, each time opening your front knee wider to the point that your front shin is parallel to the front of your mat. Focus your awareness on where you feel the stretch and breathe into it or simply focus your awareness on the center of the pain. Relax the muscles that you are stretching.
In this version of pigeon yoga pose you start with the hip of your front leg hip on the floor and your shin parallel to the front of the mat. Keep the front leg hip on the floor and focus on sinking the back leg hip. As in the previous version of pigeon yoga pose you can reach your arms forwards and use them to push your ribcage and pelvis (and back leg) rearwards.
To intensify the hip stretch use your arms to push your ribcage to the side. If your right leg is forwards then push your ribcage to the left. You can try getting your left shoulder to your right foot. Then try to get your face to your right foot. Then try to get your right shoulder to your right foot.
In all cases try pushing your front knee into the floor to help keep your front hip active. You can either rest the back knee on the floor or lift it.
You could also do this variation of pigeon yoga pose with the torso upright. With the front leg hip on the floor push the knee down into the floor. Use your front-leg arm to help push your torso to the side. Lift and straighten the back knee. Actively bend your spine backwards.