My understanding is that moves are broken down into four stages and that any ¡®move¡¯ can be broken down an infinite number of times. When practicing push hands your partner can change any number of times in any direction. Think of someone pushing down with raw, brute muscle strength on your wrists as you lift your arms from your sides in the opening move. If you do it well he may not be able to react before you complete the motion and send him jumping back. If he does react and collect himself for a continued push on your wrists you have to start over by ¡®catching¡¯ his new push, finding your direction, etc.
There are different ways of practicing with different results. Wang Yongquan writes about this specifically in his book as well. He distinguishes the ¡®for health¡¯ practice in which movements should be large, open, and have internal expansion, from the martial type of practice which has smaller, more detailed movements/changes. The ¡®for health¡¯ practice is the foundation for the martial type of practice. I think that by doing things large, open, and expanded makes clear the changes and wavelike motions inside the body. When these motions are clear one can do them smaller, quickly and in rapid succession without losing the total body connection.
So, to try to answer your question, I think it is definitely best to practice with the internal changes of the four stages corresponding to the beginning and end points of the external motions of the form. Once you feel the waves of internal movement and feel that you can change direction of large motions without losing connection you will naturally start to break things up smaller and more quickly.
Your comment below is important:
¡°I understand the Taiji energies to be analogous to the geometric formulas for straight lines, parabolas, hyperbolas, etc. Although these distinctions are meaningless at a single point of movement (i.e., Wuji), the instant there is any contrast or movement (i.e., Taiji), the nature of the curve is manifest in its entirety¡±
I think this and the other comments in this thread about curve in linear motion and lines in arcs may be clarified by thinking of a Taiji motions/energies being expressed as rotating spheres and flat circles whose centers move from point A to point B. At the same time the size of the rotating sphere or circle can expand and contract.
The word ¨Cna2- in this usage is best understood as both ¡®to pick up¡¯ (¡®something off the ground¡¯) and ¡®to take¡¯ in the sense of, ¡°it was yours and now that I have ¡®taken it,¡¯ it is mine.¡± This usage is standard Taiji terminology that occurs in the Classics in the phrases,
¡®na2 er2 hou4 fa1¡¯ (pick up and then emit)
¡®xu4 er2 hou4 fa1¡¯ (store up and then emit)
¡®xu4¡¯ is to store up as potential energy in a bow. It is the word used for charging a battery.
The object of ¡®xu4¡¯ and ¡®fa1¡¯ is, of course, ¡®jin4¡¯. Naturally, we might think that ¡®energy¡¯ would then be a logical choice for a translation of ¡®jin4¡¯. Based on my experience, ¡®jin4¡¯ needs a more concrete translation that relates directly to the elasticiy of the bow or rubber band itself and not the potential energy that the stretching of the bow creates. This deserves an entirely separate discussion altogether and is extremely important. All of the Taiji classics state very explicitly and in some detail that the process of understanding what ¡®jin4¡¯ is, is ¨Cthe- key to being able to begin practicing and understanding Taiji.
Regarding your question about pulling your jin4 back, I think that if you can experience a feeling of sending jin4 into your partner¡¯s body then you can try to do just the opposite- bring it back. When you bring it back, often your partner will follow you. If he does then you can continue to lead him out. Leading him out is called ¡®yin3¡¯ (lure) and ¡®taking up his jin4 or li4 (raw muscle strength)¡¯ is called ¡®na2¡¯. Giving it back to him is called ¡®fa1¡¯.
In Taiji it is important to always try to discover the opposite of every feeling. Try starting off with finding the opposite of the feeling of opposing gravity by standing up. Which muscles are being used to fight gravity?
Sending jin4 into the partner¡¯s body is like connecting with the gravity pulling your partner to the earth; the gravity that he is having to fight to stand erect. I think it is best to describe this as being done with varying amounts of your gravity, not your push against your gravity. Once you connect with his gravity you get a clear view or, as you say, ¡®sonar echo¡¯. After connecting with his gravity ¡®issues¡¯ you come back because you already have this clear picture. When you take your gravity back, no matter how small an amount went over, often your partner will follow your motion. If he does not follow, you can just sit and wait or you can use varying amounts of your push against gravity to trick him into pushing against his gravity.